## Tuesday, June 23, 2020

### Number Facts for Every Year Date - 181-210

The 181st Day of the Year

181 is the 42nd prime number, and the largest year day which is a strobogrammatic prime, and the third strobogrammatic prime. It is also the last strobogrammatic number of the year dates..

181 is the 9th palindromic prime number. the tenth palindrome prime is not for away, it's the next prime, 191.  *Prime Curios adds that this is the smallest pair of consecutive primes that are consecutive, and also the smallest example of two consecutive primes with no prime digits.  181 is also a palindrome when written in base 12(131)

111, also a  strobogram and is related to 181 in that 111!  has 181 digits *Prime Curios

If you square 181 and add 7, you get 32768. So what? Well 32768 is 215.  STILL not impressed? The only other numbers for which n2 + 7 is a power of 2 are 1, 3, 5, and 11.... full stop. And to take this beyond the coincidental, if you replace the 7 with any other integer, there will never be more than two solutions. The problem was posed by Ranujan in 1915,  who asked were there any others than these five.

181 is the sum of 23 consecutive primes, 2+3+5+...+79 + 83 =181 *Prime Curios and also the sum of fiv e consecutive primes 29+31+37+41+43= 181

*Prime Curios has "The smallest palindromic prime that remains prime through four iterations of the function f(x) = 2x + 5." leaving the suggestion that there are other smaller primes that remain prime through four iteration of f(x) = 2x+5. 7 works three times, but not the fourth. You have to go just a little higher.

and the 181-digit palindromic number made up of all 7's except for the center being 181 (7777...7718177...77777) is a palindromic prime with a palindromic prime decimal length.

181 is the both the difference and the sum of consecutive squares:
$$181 = 91^2 – 90^2 = 9^2 + 10^2$$

181 is the fifth prime (but the first palindromic prime) for which 2^p bisects the gap between two primes, $$2 ^ {181} \pm 165$$ are two consecutive primes. *Prime Curios

181 is an Emirp with itself, and the sum of three emirps, 31+71+73 = 181.

181 is a centered pentagonal number, the sequence of pentagonal numbers begins 1, 6, 16,  31, 51..
 *Wikipedia

Every natural number greater than 181 can be written as sum of cubes of the first two primes. (students might be asked to find all examples of numbers less than 181 that can be written in this fashion, such as 35= 23 + 33)

The 182nd Day of the Year

The 182nd prime (1091) is the smaller of a pair of twin primes (the 40th pair, actually) *Math Year-Round ‏@MathYearRound(Students might convince themselves that it was not necessary to say it was the smaller of the pair.)

While literally every small (less than 945) odd number is deficient, 182 is the 91st even number, and only the 48th even number to be deficient. In all then up to 182, there are 182 deficient numbers, and only 43 that are abundant. "The natural numbers were first classified as either deficient, perfect or abundant by Nicomachus in his Introductio Arithmetica (circa 100 CE)".*Wikipedia

Language time:
182= 13*14  is called a pronic, promic, or heteromecic and even an oblong number. Pronic Numbers are numbers that are the product of two consecutive integers; 2, 6, 12, 20, ..(doubles of triangular numbers).  Pronic seems to be a misspelling of promic, from the Greek promekes, for rectangular, oblate or oblong. Neither pronic nor promic seems to appear in most modern dictionaries. Richard Guy pointed out to the Hyacinthos newsgroup that pronic had been used by Euler in series one, volume fifteen of his Opera, so the mathematical use of the "n" form has a long history. Oblong is from the Latin ob (excessive) + longus (long). The word oblong is also commonly used as an alternate name for a rectangle. In his translation of Euclid's "Elements", Sir Thomas Heath translates the Greek word eteromhkes[hetero mekes - literally "different lengths"] in Book one, Definition 22 as oblong. . "Of Quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right angled but not equilateral...". (note that with this definition, a square is not a subset of rectangles.)
As the product of three distinct primes , 182= 2*7*13, it is also a sphenic or wedge number

182 is the smallest pronic number (not ending in zero) whose reversal is a prime.

182 is a palindrome in base 3 (20202) and a palindrome and repdigit in base 9 (222)

The regular polygon with 182 sides, has exterior angles at each vertex of less than 2 degree. Coxeter called all these evenly sided, 2*n,  polygons zonagons and said that they could be divided into n(n-1)/2 parallelograms, and in the case of regular polygons, they will all be rhoumbi (but not all identical rhombi), so the 2*91 = 182 sided zonagon will have 91*45=4095 rhombi (too many to make a good image, so here is an Octadecagon with only 36 from the nice people at Wikipedia)  (these disections can be done in a multitude of ways, so I just picked a pretty one).

The 183rd Day of the Year
the concatenation of 183 and 184, 183184 is a perfect square. There are no smaller numbers for which the concatenation of two consecutive numbers is square. (Students might seek the next such pair of numbers. They are small enough to be year dates)

In the toothpick sequence, at the 18th level, there are 183 toothpicks.  "In geometry, the toothpick sequence is a sequence of 2-dimensional patterns which can be formed by repeatedly adding line segments ("toothpicks") to the previous pattern in the sequence.
The first stage of the design is a single "toothpick", or line segment. Each stage after the first is formed by taking the previous design and, for every exposed toothpick end, placing another toothpick centered at a right angle on that end"*WIkipedia  The sequence begins 1, 3, 7, 11, 15, 23.. it forms a fractal shape. This Wikipedia gif shows the first three steps.
The image after 89 steps looks like this:

 *WIkipedia

The sum of the first 183 primes minus 183 is prime.

183 is the eighth of the 12 year-days which are perfect totient numbers. (There are only 57 such numbers under 103). A list of the perfect totient numbers seems to suggest that all of them are multiples of three, but then you get to 4375, the smallest perfect totient number that is not divisible by 3.[a perfect totient number is a number that is the sum of it's iterated totients, that is, the number of integers smaller than, and relatively prime to 183 + the number smaller than and less than that result, + ... down to one, "For example, start with 327. Then φ(327) = 216, φ(216) = 72, φ(72) = 24, φ(24) = 8, φ(8) = 4, φ(4) = 2, φ(2) = 1, and 216 + 72 + 24 + 8 + 4 + 2 + 1 = 327 " *Wik
If you multiply every digit of a standard 3x3 magic square by 12, you get a 3x3 magic square with a sum of 180, but if you add one more to each entry....
49   109   25
37    61   84
97    13   73

183 is the difference of two squares, 32^2 - 29^2, and of course, like every odd number, it is the difference of the squares ot the consecutive numbers that sum to 183, 92^2-91^2= 183

Lagrange proved that every integer is the sum of four or less non-zero squares.  183 is one of the unusual ones that require four. It is the 29th number that requires the full set of three squares. $$183 = 13^2 + 3^2 + 2^2 + 1^1.$$

The 184th Day of the Year
The 184th day of the year; 184 = 23 * 23 (concatenation of the first two primes).

The smallest number that can be written as q * pq + r * p r, where p, q and r are distinct primes (184 = 3 * 23 + 5 * 25). *Prime Curios

184 is the sum of four consecutive prime numbers 41+43+47+53 = 184

184 is a balanced number in binary, with equal numbers of zeros and ones.  It is the 14th such number in the year so far.

184^2 + 1 =33857 is a prime

The concatenation of 183 and 184, 183184 is a perfect square. There are no smaller numbers for which the concatenation of two consecutive numbers is square. (Students might seek the next such pair of numbers. They are small enough to be year dates)

On a 5x5 lattice (square grid of dots) there are 184 paths from one corner to the opposite corner touching each lattice point exactly once.

184 is a refactorable or Tau number, it's divisible by the count of its divisors. It has eight divisors, including 8. (1, 2, 4, 8, 23, 46, 92, 184).,

184 is the difference of two squares 25^2 - 21^2  aand the sum of three squares 12^2 + 6^2 + 2^2  and of four squares, 10^2 + 8^2+4^2 + 2^2

The 185th Day of the Year
The 185th day of the year; the decimal expansion of the first 185 digits of Euler's constant is prime. *Prime Curios

Numbers ending in five, like 185, are always the difference of squares that are five apart, in this case 21^2 - 16^2 =185, and of course, every odd number is the difference of two consecutive squares, 93^2 - 92^2.

185 is a semiprime, the product of two distinct primes, 5*37.

185 is the sum of two square numbers in two different ways: $$13^2+ 4^2$$ and $$11^2 + 8^2$$  I'm not sure it is commonly known that this implies that these pairs can be used as the opposite sides of a quadrilateral forcing the diagonals to be perpendicular.  (if the sides of a quadrilateral are 13, 11, 4, 8; then the quadrilateral has perpendicular diagonals.)

That also means that 185 is the hypotenuse of four Pythagorean Triangles,
(60, 75, 185) (111, 148, 1854)(57, 176, 185) (104,153,185)

185 =the sum of five squares, 100+64+16+4+1

185 is a palindrome in base 6(505) 5*6^2 + 5.

The 186th Day of the Year
There are 186 days between the Spring and Fall Equinox, which is well over 1/2 a year. The reason, we are on the wrong side of the Earth's Elliptic orbit and have to travel a greater distance. From Fall to Spring takes only 179 days. (there is of course, an extra quarter of a day in there somewhere.)

186 is the product of the first four primes less; the product of the first four positive integers, 7# - 4! (7 x 5 x 3 x 2 - 4 x 3 x 2 x 1 = 186) . *Prime Curios Students might not have seen the p# symbol, it represents the Primorial, the product of all the primes from p down to 2.

186 is the sum of consecutive primes, 186 = = 89 + 97,

186 is a sphenic (wedge) number, product of 3 distinct primes: 186 = 2*3*31

Another number with a nice palindrome expressions, 3*3*3 + 3+13 + 31*3 + 3*3*3 (easy as 1,2,3, but without the 2)

and from Jim Wilder @wilderlab An equation for July 4th: 7⁴ = 2401 (2 + 4 + 0 + 1 )4 And a follow up from World Observer@WKryst2011 points out that there are only two other such year dates. (student's should find both)

186 is a palindrome in base 5(1221), and in base 8(272)

The 187th Day of the Year
187^(1*8*7)+1+8+7 is prime. There are only two such (non-zero) numbers. Students might search for the other.

The 187th prime is 1117. 11*17 = 187

187² and 187³ don't have 1, 7, or 8. *Math Year-Round ‏@MathYearRound

With 187 people in a room, there's a 50% chance that 4 share the same birthday *Derek Orr

187 is the sum of three consecutive primes 59, 61, and 67 ; but also the sum of nine consecutive primes starting at 7, and ending at 37.

Sesame Street Muppet Count von Count's favorite number, 34969, is 187x187 (1872), speculated in a BBC mathematics program to be a reference between Count Dracula and the Californian murder code (sect 187 of the California Criminal code, sometimes used as a slang word for murder).Sesame Street Muppet Count von Count's favorite number, 34969, is 187x187 (1872), speculated in a BBC mathematics program to be a reference between Count Dracula and the Californian murder code.

187 is hexdecimal (base 16) is a tiny number, as small as a BB (there is a joke hidden in there somewhere.)

Every odd number is the difference of the squares of two numbers that differ by one and sum to p, in this case 197 = 99^2-98^2,but 197 is also the difference of two other squares, 14^2 - 3^2

Euler's Pentagonal number theorem is important in the computation of the number of partitions of a number. They alter the Pentagonal number forumla $$p_n = \frac{3n^2 - n}{2}$$ which generates the Pentagonal numbers when n is a positive integer, by including after each positive integer, it's opposite. The function values at each of these successive terms become the exponents of the variable in his Pentagonal theorem. I mention this here, of course, because the exponenet of the 22nd term is 187.

The 188th Day of the Year
188 is the sum of six distinct squares. 1 + 4 + 9 + 25 + 49 + 100 = 188. Any larger number can be formed with no more than five distinct squares.

188 is the largest known even number that can be expressed as the sum of two (distinct) primes in exactly five ways. *Prime Curios  Students might seek smaller numbers that can be so expressed..

Neither 1882 nor 1883 contain a one or an eight. *@Derektionary

There are 188 11 bead necklaces using two colors, if the necklace can not be turned over.

188 is a Happy number: trajectory under iteration of sum of squares of digits map to 1.

The largest known even number that can be expressed as the sum of two (distinct) primes in exactly five ways. *Pimre Curios

188 is a product of 4 times a number (47). Any such number is the difference of two squares, one of which is the square of one more than the number n/4, and one of which is the square of one less. 48^2-46^2 = 188

The immortal Casey Jones of country music ballads was a real guy (and born in the town of Cacey in Fulton County, Ky) and on April 30, 1900 he took off from Jackson, Tennessee bound for Canton, Mississippi on the Cannonball, but was killed in a dark foggy night when a stranded train was on his rail in Vaughn, Mississippi. His skilled driving saved his passengers, but his life ended at mile number 188 of his final drive.

The 189th Day of the Year
the product of the primes in a prime quadruplet always end in 189, except for the very first quadruplet 3x5x7x11.(A prime quadruplet (sometimes called prime quadruple) is a set of four primes of the form {p, p+2, p+6, p+8} you can see some of the smaller ones listed here

There are 14 prime years in the 21st Century (2017 will be the third), but the 189th century would be the first to contain as few as five prime years (18803, 18839, 18859, 18869 and 18899).

Narayana, an Indian mathematician in the 14th century, came up with an interesting Fibonacci-like series: A cow produces one calf every year. Beginning in its fourth year, each calf produces one calf at the beginning of each year. How many cows and calves are there altogether after n years? For the 15th year, the total is 189. (How many mature and immature?)

2357 is a prime number. 23357 is also prime.  233357 is also prime but 2333357 is not, and then 23333357 is; and yes, this is somehow related to the number 189. I came across a sequence on OEIS  which gave "Numbers k such that (7*10^k + 71)/3 is prime." Like you may have, I wondered, "Why would someone search for primes of so unusual a sequence?" Well, if you take those prime numbers, and subtract 2 from them, you get the number of threes that when placed between the digit 2 and the digits 57, will produce a prime. So I can inform you today that not only is (7*10189 +71 )/3 a prime number, but that prime number is a 2 followed by 187 threes followed by 57.  And you thought 189 was just some hum-drum number!!!!!

189 = 13^2 + 4^2 + 2^2 and

189 = 1+89 + 98 + 1 a palindrome using only the digits of the number twice each.

$$189 = 95^2 - 94^2$$ and of course, as previously mentioned, it is 6^3 - 3^3; and  $$189 =5^3 + 4^3 = 15^2 - 6^2= 17^2-10^2$$

The 190th Day of the Year
The 190th day of the year; 190 is the largest number with only distinct prime Roman numeral palindrome factors that is a Roman numeral palindrome (190 = CXC = II * V * XIX). *Prime Curios

190 is also a palindrome in base 4(2332) 190 is a Harshad or Niven number divisible by the sum of its digits. In recreational mathematics, a Harshad number (or Niven number) in a given number base, is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers in base n are also known as n-Harshad (or n-Niven) numbers. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "Harshad" comes from the Sanskrit harṣa (joy) + da (give), meaning joy-giver.

190 is the 19th triangular number, the sum of the first 19 integers. A nice problem relating triangular numbers to magic squares was asked in 1941 in the American Mathematical Monthly, posed by Royal Vale Heath, widely known for creating ingenious mathematical puzzles: "What is the smallest value of n for which the n2 triangular numbers 0, 1, 3, 6, 10, . . . n2(n2 – 1)/2 can be arranged to form a magic square?" An explanation, and answer is in this blog by Ivars Peterson

190 is a Happy Number. Summing the squares of the digits, and iterating, you eventually arrive at 1. It takes only four iterations.

190 = 121 + 49 + 16 +4 = 100+81+9

The 191st day of the Year
191 is a palindromic prime and when it is doubled and one is added to this result, the resulting number is yet another palindromic prime. (Students might consider why 11 is the only palindromic prime with an even number of digits.)

Derek Orr noticed that 199*n+(n-1) is a palindrome (not prime) for several other values of n, collect the whole set.

By adding up the values of the common US coins, one obtains 191 ¢ (silver dollar + half dollar + quarter + dime + nickel + penny) *From Number Gossip (This ignores the once minted 5 mil, or half-cent coin and the briefly lived 2 cent coins) Canadians would have a larger sum of coins since Canada has had a $1 coin (The Loonie) since 1987 and a$2 coin (The Toonie) for about 10 years. I think the Canadian total would be 341 (no  half dollar) so maybe  we can squeeze them in by the end of the year.

191 is the smallest palindromic prime p such that neither 6p - 1 nor 6p + 1 is prime. Also, The smallest multidigit palindromic prime that yields a palindrome when multiplied by the next prime: 191 * 193 = 36863. *Prime Curios

191 is in the lazy caterer sequence, since 19 slices of a pie can produce 191 pieces.  19 is related to 191 in another way, One Hundred Nighty-One, count the digits.

191 is formed by three square numbers, 191^2 = 36481, a square (4) with a square in front of it (36) and behind it (81)

191 is a palindrome in base 6 (515)

191 is the first prime in a prime quadruplet, 191, 193, 197, 199.  The sum of their digits are also prime 11, 13, 17, and 19.

The 192nd Day of the Year
The 192nd day of the year; 192 is the smallest number that together with its double and triple contain every digit from 1-9 exactly once. There are three other values of n so that n, 2n, and 3n contain each non-zero digit exactly once. Can you find them?

192 is the sum of ten consecutive primes (5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37)

192 is the number of edges on a 6th dimension hypercube, it is the last day of the year which is the number of edges of a hypercube.

192 is a Happy number, summing the s1uare of its digits and iterating leads to 1 in only three iterations. Its also a Hashard (Joy-giver) number, divisible by the sum of it's digits.

Diophantus probably knew, and Lagrange proved, that every positive integer can be written as a sum of four perfect squares. Jacobi] proved the stronger result that the number of ways in which a positive integer can be so written equals 8 times the sum of its divisors that are not multiples of 4. Use this theorem to prove that there are 192 ways to express 14 as a sum of four squares.

Because 192 is divisible by four, it is the difference of two squares n^2 and (n+2)^2, 49^2 - 47^2; because it is divisible by 8, it is the difference of two squares that n^2 + (n+4)^2; 26^2 - 22^2. But don't stop there, because it is divisible by 16, it is the difference of two squares n^2 and (n+6)^2; 19^2 - 13^2 , and YES, it it also divisible by 32, so one more time, 16^2-8^2.

192 = 3*4^3 and also one more than 3 * 8^2

192 is called a practical number, because you can make a subset of its divisors to sum to any number less than 192.

The 193rd day of the Year
193 = 12^2 + 7^2
193 is also 97^2  - 96^2

193 is one of a twin prime pair and is the sum of products of the first three twin primes pairs: 3*5 + 5*7 + 11*13 = 193. *Prime Curios The square of 193 (37249) concatenated with its reverse (which is a prime) results in a palindrome (3724994273) that is the product of 2 palindromes, one non-prime (1001) and one prime (3721273). *Prime Curios

The product of the first four primes, 2*3*5*7 = 210, their sum is 17, the product minus the difference is 193.

193 is a happy number, if you sum the square of the digits, and do the same with each answer, you will eventually wind up with 1.

Like all odd numbers, it is the difference of two consecutive squares, 97^2- 96^2 and 96+97 = 193.

193 is a palindrome in base 12 (121) so 193 = 12^2 + 2*12 + 1

193 is one more than 3*4^3 and also one more than 3 * 8^2 193 is the smallest prime whose fifth power contains all digits from 1 to 9.

(I also like 193/71 is the closest ratio of two primes less than 2000 to the number e.)

The 194th Day of the Year
1944+1 = 1,416,468,497 is prime *Prime Curios 194 is also the smallest number that can be written as the sum of 3 squares (not all unique) in five ways. (There is a slightly larger number that is expressible as the sum of 3 unique squares in five ways. )

194 = 13^2  + 5^2

194 is the smallest Markov number that is neither a Fibonacci number or a Pell Number, Markov nuimbers are any of the x, y, z, such that x^2 + y^2 + z^2 = 3xyz, and in this case they are 5, 13, 194.  ( I admit I have some work to do to figure out what makes these triples special and the chains on which they are formed.  )

194 is a palindrome in base 3, (21012), and in base 6 it is (1234) which is cool.

194 is the product of the largest and smallest prime less than 100.

194 is the sum of three consecutive squares, $$194 = 7^2 + 8^2 + 9^2$$

194 is the even base of the Largest Heronian triangle with consecutive integer sides that can be year dates.  Heronian triangles are triangles that have all three sides and the area as integers.  The three sides are 193, 194, 195, and the Area is 16,296 sq units.  I have seen these called Super Heronian triangles, but I call them Sang-Heronian triangles after the earliest study I know about them by Edward Sang of Edinburgh, Scotland in1864  Because these bases are always equal, the altitude from that base must also be an integer. And one more biggie... If you construct the altitude to the even base, one side or the other of it will always form a primitive Pythagorean triangle.  For each new bigger triangle, it switches sides.  In the triangle for this date, the PPT is 95, 193, 195.  I won't to write a little more about this than space here allows, so I will link it here then.

The 195th Day of the Year
Take the basic 3x3 magic square, multiply by 13, and get<br>
52 117 26
39   65 91
104 13 78
A magic square with a constant of 195.  13*15=195  different multiple gives you constants of 15k for any k.

195 is a palindrome in binary (11000011)  and as you can see, it is a balanced number with exactly the same number of zeros and ones. It is also a palindrome in base 4.  Divided the digits into groups of two and convert their value to decimals, 11=3, 00=0, 00=0, and 11=3 so in base four, it is (3003), and in base 8 (303) by taking three digits at a time (right to left)  11,   000,   011

If you take the basic 3x3 magic square with digits from 1 to 25, and multiply each term by 3, you get a 5x5 magic square with a magic constant for each row and column of 195
51   72   3   24   45
69   15   21   42  48
12   18   39   60  66
30   36   57   63  9
33   54   75   6    27

if you keep 39 in the middle and replace all numbers in their order by incrementing by one, you get another magic square for 195.  (42 will be 40, 36 will be 38, etc.)

Add one to each term and you have a magic square for day 200!

195 is in the sequence of integers 6n+9 for n = 31 and so 195 = 34^2 = 31^2
and because it is in 10n+25 for n=17, 195 = 22^2 - 17^2, and like every odd number, 195 = 98^2 -97^2, , the difference of the squares of two consecutive numbers that add up to 195.

195 is the sum of eleven consecutive primes: 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 Students might wonder which numbers can (and cannot) be expressed as the sum of one or more consecutive Primes.

195=5^2 + 7^2 + 9^2  It is only  the third integer that is the sum of the squares of three consecutive primes.*Prime Curios  5
There are lots more ways to find distinct squares that sum to 195, as it is the smallest number expressed as a sum of distinct squares in 16 different ways. *Wik

also, 1*95 = 19*5 Derek Orr tells me there are only four non-trivial 3-digit numbers with this property *@Derektionary

a Heronian triangle is a triangle that has side lengths and area that are all integers. There is an almost-equilateral, scalene triangle with one side of 195. The other sides are 194, and 193. Students can find the area using Heron's formula.

The 196th Day of the Year
14^2 = 196;

A Lychrel number is a natural number which cannot form a palindromic number through the iterative process of repeatedly reversing its base 10 digits and adding the resulting numbers. 196 is the lowest number conjectured to be a Lychrel number; the process has been carried out for one billion iterations without finding a palindrome, but no one has ever proven that it will never produce one. The number produced on the one billionth iteration had 413,930,770 digits. The name "Lychrel" was coined by Wade VanLandingham—a rough anagram of his girlfriend's name Cheryl. No Lychrel numbers are known, though many numbers are suspected Lychrels, the smallest being 196. (Students might try finding the number of iterations of the process to find a palindrome for various n. 195, for example, takes four iterations : 195 + 591 = 786 786 + 687 = 1473 1473 + 3741 = 5214 5214 + 4125 = 9339) DO not try the numbers 89 or 98. Harry J Saal used a computer to repeatedly iterate this process and finally did come up with a palindrome, the number 8,813,200,023,188 on the 24th iteration.

Jim Wilder noticed that 142 =196 and 132=169... are there other squares of consecutive numbers that share the same digits?

196 is the aliquot sum of 140, 176 and 386

196 is a palindrome in base 13, (212) (I never do had much luck working in base 13)

and a palindromic expression of 196 useing only its digits, 19 + 16 +  9 + 61 + 91  OR 96 +1 + 9 + 11 + 9 + 1 + 69
A number is said to be square-full if for every prime, p, that divides it, p2 also divides it. 196 is such a number, 196 = 2^2 + 7^2   Are there cube-full numbers? (of course there are, but what are they? 8 would be, as would any cube, I guess smallest with more than one is 6^3 = 216, coming up soon)

Because 196/4 = 49, it must be the difference of two squares, 50^2-48^2.,

The 197th Day of the Year

197 is the sum of all digits of all two-digit prime numbers: 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. It is simple to show that the sum of one-digit primes is 17. Do the sum of the digits of n-digit primes always end in seven? (http://oeis.org/A130817) Or perhaps we ask, are there any others that do?

196 was a square-full number, and 197 is a square free number, since it is the 43rd prime

197 is the smallest prime number that is the sum of 7 consecutive primes: 17 + 19 + 23 + 29 + 31 + 37 + 41 (student challenge: can there be a prime that is the sum of eight consecutive primes?)

197 is the sum of the first twelve prime numbers: 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37, and obviously the smallest number to be the sum of twelve primes. (You have to wait 39 days for the next one.

197 is the last year date which is a Keith number . The Keith numbers is a digit summing process in which one digit is replaced by the new sum. For example, since 197 is a three digit number, we will add three digits in each iteration. We start with the digits of the number, 1+9+7 = 17. Now we delete the first digit, 1, and add the 9+7 +17 = 33, Again, delete the first of the three numbers, 9 and add 7 + 17 + 33 = 57. Continue the process 17+33+57= 107, and once more for; this one, 33+57+107 = 197, back to the number we started with. Such numbers are called Keith Numbers after Mike Keith, an American mathematician working at Sarnoff created the idea in 1987. There are only about 100 known in base ten, and only 7 that are year dates, 14, 19, 28, 47, 61, 75, and 197. Keith used the term repfigit, and that name is still used by some today.

197 = 111 + 9 +77 sum of three repdigits

197 = 14^2 + 1^2 and 197 = 99^2-98^2

The 198th Day of the Year;
198 is a Harshad number, divisible by the sum of its digits. A Harshad number, or Niven number in a given number base, is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "Harshad" comes from the Sanskrit harṣa (joy) + da (give), meaning joy-giver. The Niven numbers take their name from Ivan M. Niven from a paper delivered at a conference on number theory in 1997. (Students might try to find a pair of consecutive numbers greater than 10 which are harshad numbers)

198 nines followed by a one is prime 9999...... 91. *Derek Orr@Derektionary

198 is between the twin primes 197 and 199.

If you multiply 198 by its reversal, 891, you get 176,418 which is between the twin primes 176,417 and 176,419. Is there another example of this curiosity?

198 = 13^2 + 5^2 + 2^2

Palindrome expressions for 198 = 2 x 72 + 27 x 2 = 3 x 15 + 51 x 3 = 55 + 88+ 55

19 is the 8th prime number, and if you concatenate them, 198 is the (19*8= 152nd) composite number.

The difference between any two Emirp pairs is divisible by 198 *Prime Curios

198 is called a practical number because every number from 1 to 197 can be written as sums of divisors of 198.

The 199th Day of the Year
199 is prime (in fact, all three permutations of the number are prime) and is the sum of three consecutive primes: 61 + 67 + 71, and of five consecutive primes: 31 + 37 + 41 + 43 + 47. (Suddenly struck me I don't know what is the smallest prime that is the sum of consecutive primes in more than one way!)

199 is the sum of the digits of all the three-digit palindromic primes.  *Prime Curios <br><br>
199 is the smallest number with an additive persistence of 3. (iterate the sum of the digits. The number of additions required to obtain a single digit from a number n is called the additive persistence of n, and the digit obtained is called the digital root of n. ) 1+9+9 =19, 1+9=10, 1+0 = 1. so the additive persistence is 3 and the digital root is 1.

I like "almost constants". For the 199th day, $$( \frac{\sqrt{5} +1}{2})^{11}= 199.0050249987406414902082…$$

199 is the last year day that is part of a prime quadruplet, (191, 193, 197, 199)

199 is the smallest number that has an additive persistence of 3, 1+9+9 = 19; 1+9 =10; 1+0=3 *Prime Curios

199 = 100^2-99^2

199 as a palindrome of its own digits, 99+1+99=199= 9*9+9*1+9+1+9+1*9+9*9

199 is a permutable prime, and 919 and 991 are both prime

199 is the first prime number in a sequence of 10 consecutive prime numbers with common difference 210 (tao and green 2008; see R.Taschner "Die Farben der Quadratzahlen" p. 147)the ten primes are 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, and 2089.

The next prime after 199 is 211. If they are concatenated in either order, they form a prime, 199211 and 211199 are both prime. *Prime Curios

199 is the smallest emirp that is also an invertible prime, it's 180 degree rotation produces the prime 661. *Prime Curios

199, 211, and 223 are the smallest triple of primes of the form n, n+12 and n+24, and it is the only triple less than 1000. *Prime Curios

The 200th Day of the Year
200=14^2+2^2 and of course 10^2+10^2.

200=51^2-49^2=27^2-23^2

200 is the smallest unprimeable number - it can not be turned into a prime number by changing just one of its digits to any other digit. (What would be the next one? {easier}What is the smallest odd unprimeable number? {harder})

Sum of first 200 primes divides product of first 200 primes. (How often is this property true of integers?) *Math Year-Round ‏@MathYearRound

The smallest discernible movement of a computer mouse—equal to 1/200th of an inch—is called a MICKEY. Haggard Hawks @HaggardHawks The actual measure depends on the equipment of course, so, like the meter, it will have to be adjusted from time to time.

A 5x5 magic square for 200 is easy.  Take the standard 5x5 square using 1-25, multiply all entries by 3x+1. ,BR.,BR.

51  72    4  25  46 <br>
70  16  22  43  49<br>
13  19  40  61  67<br>
31  37  58  64  10<br>
34  55  76   7  28<br>
<br><br>
And don't forget, When you pass GO, collect $200 (except the lucky Birts, who get 200 Lb.<br><br> It is not a palindrome in any base 2-10, but it is in Roman numerals CC. <br> Which reminds us that 200 is the sum of two squares, 10^2 + 10^2, but also 14^2 + 4^2, <br>And as the difference of two squares, 200 = 51^2 - 49^2; 27^2 - 23^2; 15^2 - 5^2 The 201 Day of the Year 201 is a harshad number... A Harshad number, or Niven number in a given number base, is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "Harshad" comes from the Sanskrit harṣa (joy) + da (give), meaning joy-giver. The Niven numbers take their name from Ivan M. Niven from a paper delivered at a conference on number theory in 1997. (Can you find the string of three consecutive Harshad numbers smaller than 201?) 201 is also a lucky number, a number that survives from the sieve process created about 1955 by Stanislaw Ulam, the great Polish mathematician who coinvented the H-bomb and was the father of cellular automata theory. Students who are familiar with the way the Sieve of Erathosthenes produces the primes may wish to compare the lucky numbers produced by this sieve. "Start wtih the odd numbers.The first odd number >1 is 3, so strike out every third number from the list (crossing out the 5, 11,17 etc): 1, 3, 7, 9, 13, 15, 19, .... The first odd number greater than 3 in the list is 7, so strike out every seventh number: 1, 3, 7, 9, 13, 15, 21, 25, 31, .... The numbers that remain are the so called "lucky numbers". Look for similarities to the primes. *Martin Gardner, Mathworld 201= 101^2-100^2 = 35^2-32^2 201 has a largest factor of 67, so a palindrome of 201 with only 6 and 7, 6 x 7 + 7 x 6 + 7 + 6 + 7 + 6 + 7 + 6 x 7 + 7 x 6 = 201 201 is a Joy-Giver or Harshad number, divisible by the sum of its digits 201 is the the difference of two squares, 201 = 101^2 - 100^2=35^2 - 32^2, and just to break the rules, it's also 22.6^2 - 17.6^2 And Wikipedia tells me that Star Trek had an episode with the title, 11001001, which is 201 in binary. A 3x3 magic square with a magic constant for each row of 201 can be created by taking the basic 1-9 digits and replacing them with 13n+2 54 119 28 41 67 93 106 15 80 The 202nd Day of the Year in an alphabetical listing of the first one-thousand numbers, 202 is last. 202293 begins with the digits 293 and 293202 begins with the digits 202. *jim wilder ‏@wilderlab To get a fill of how rare it would be for this kind of reversal, students might search for just numbers that work only one way. For example 6^13 starts with a 13, but the reversal doesn't work. Sounds like a good idea for a computer search; if you find more, send me a note. There are 46 palindromes in the 365 (or 366) days of the year, 202 is the 30th of these. (2+3+5+7)2 -(2 2 + 32 + 52 + 72) =202 *Prime Curios If your digital clock uses a seven digit display, such as many microwaves, stoves, and small alarm clocks, then 202 is a strobogrammatic number, turn the clock (not the stove, please) over and it reads the same. There are exactly 202 partitions of 32 (2^5) into smaller powers of two *Wikipedia 202 in Roman numerals and read in English sounds like a "Yes" in both Spanish and Nautical, "Si, Si, Aye Aye" 202 = 11^2 + 9^2 The 203rd Day of the Year 203 = 18^2-11^2 = 102^2-101^2 203 is the 6th Bell number, i.e. it is the number of partitions of a set of size 6. 203^2 + 203^3 + 1 is prime. 203 is the number of nondegenerate triangles that can be made from rods of lengths 1,2,3,4,...,11 203 is the number of triangles pointing in opposite direction to largest triangle in triangular matchstick arrangement of side length 13 Saw a tweet about July 22 as "Casual Pi Day" at Rimwe@RimweLLC which he told me he found at page of GeorgeTakei. The NCTM uses "Pi Approximation Day" for it's poster 203 is a palindrome in base 3 (21112) base 6 (535), and base 8(313), 203 is the sum of the squares of five consecutive primes. No smaller such prime exists, and no other day number has this quality. The 204th Day of the Year 204 is the eighth tetrahedral number, the sum of the squares from 1 to 8. Answers the question, how many squares are there on an 8x8 checkerboard. 204 = 20^2 - 14^2=52^2 - 50^2& 204 is the sum of consecutive primes in two different ways: as the sum of a twin prime (101 + 103) and as the sum of six consecutive primes (23 + 29 + 31 + 37 + 41 + 43). (one might wonder what is the smallest number that is the sum of consecutive primes in more than one way... And what is the smallest prime number that is expressible as the sum of consecutive Primes in more than one way?) And a trio from *Derek Orr @MathYearRound : 204 = 1²+2²+3²+4²+5²+6²+7²+8². Sum of first 204 primes is prime. 100...00099...999 (204 0's and 204 9's) is prime. 204^n + b1 is prime when n = 2, 0, 0r 4. *Prime Curios 204 is a refactorable number, it is divisible by the count of its divisors, 12 It is because it is divisible by 12 that 204 is expressible as (n+3)^2 - (n-3)^2 for n= 17 (204/12=17) 204^2 = 41616, a number that is both a square and a triangular number. They are pretty rare as it is only the fourth. And like 204, it is the sum of twin primes also, 41616 = 20807 + 20809. Only 12 and 84 also are the sum of twin primes with a square that is also the sum of twin primes. since I've written twin primes so many times in this entry, its probably a good time to remind you that, although the idea of primes seperated by only a single composite number was known back to antiquity, the term was created around the end of the 19th century by German Mathematician Paul Stackel (in the German, of course, "Primezahlzwilling") On an infinite chess board, a Knight can reach 204 different squares in eight moves. 200 is CC in Roman numerals, and 204 is CC in hexadecimal. The 205th Day of the Year there are 205 pairs of twin primes less than ten thousand. *Number Gossip Every number greater than 205 is the sum of distinct primes of the form 6n + 1. *Prime Curios 205 is the number of walks of length 5 between any two distinct vertices of the complete graph K_5 205 = 14^2 + 3^2 ;= 13^2+6^2 205 = 103^2 - 102^2 = 23^2 - 18^2 The first is from a property of every odd number, the second is from a property of any number greater than 35 that ends in five. 205 = 2 x 41 + 41 + 41 x 2 also 54 + 45 + 7 + 54 + 45. or 99 + 7 + 99 205 is a palindrome in duodecimal (base 12, (151) =1 x 12^2 + 5 x 12 + 1) The 206th Day of the Year 206 is the lowest positive integer (when written in English) to employ all of the vowels once only. (This seems to require the use "two hundred AND six" which I really dislike. What would be, or is there a, first without this "and"?) (Michael King ‏@processr suggested "5000 fIvE thOUsAnd". 206 is Sum of the lengths of the first runs in all permutations of [1, 2, 3, 4, 5] (for example, the first run of the permutation 23541 is three.) 206 is the sum of 39, and the 39th prime, 167. There are 206 bones in the typical adult human body. (I suppose all those spineless people folks talk about have well under 200!) 206 is the 36th year date for which n^2 + 1 is prime. It's also the 59th day for which n^2 + n + 1 is prime. Am I the only one who thought the second would be more unusual? 204, 206 and 208 are all the sum of a square and a cube. 206= 5^3 + 9^2. It seems that there are an infinite numbers of three consecutive integers that are such sums. 126, 127, 128 and 129 is a string of four such sums There are 206 partitions of 26 into four parts The sum of the divisors of 14 and 15 are equal. I mention that here because the next occurrence of such an incident is 206 and 207 which both sum to 312. The 207th Day of the Year 207 is the smallest possible sum of primes which are formed using each of the digits 1 through 9 (i.e., 89 + 61 + 43 + 7 + 5 + 2 = 207) *Prime Curios (So how many such sums can there be? And which of such sums are prime?) There are exactly 207 different matchstick graphs with eight edges ( a matchstick graph is a graph that can be drawn in the plane in such a way that its edges are line segments with length one that do not cross each other) Here are a few of them: 207 = 16^2 - 7^2 = 104^2 - 103^2 = 36^2-33^2 207 =3 x 3 x 23, and 3^207 + 3^207 + 23^207 is a prime. Prime Curios 207 = 9 times the 9th prime, and it is the 9th year day that is n times the nth prime. 207 = 104^2 - 103^2, and because 6 x 33 + 9 = 207, 207=36^2 - 33^2. 207 is the 33rd day of the year that are the sum of 4, but no fewer nonzero squares. The 208th Day of the Year 208 is the sum of the squares of the first five primes. 208 is the number of paths from (0,0) to (7,7) avoiding 3 or more consecutive east steps and 3 or more consecutive north steps. 208 is an abundant number, the proper divisors total 226(more than 208) 208 = 6^3 - 2^3 208 is the sum of a cube and a square, as were 204 and 206. 208 = 4^3 + 12^2 208 = 8^2 + 12^2. 208 = 53^2 - 51^2 = 17^2 - 9^2 = 28^2 - 24^2 (16*10^208-31)/3 is prime, and it has a 5 followed by 206 threes, finished of with 23. It is the largest year date in this sequence. Previous examples include 523, 5323, 53323, and 5333333333333323, for the exponents 1, 2, 3, 4 and 15 $$208 = 2^2 + 3^2 + 5^2 + 7^2 + 11^2$$, the sum of the first five squares, obviously the smallest number to be the sum of five distinct squares of primes. The 209th Day of the Year 209=16+25+34+43+52+61. Also 209 is a "Self number" A self number, Colombian number or Devlali number (after the town where he lived) is an integer which, in a given base, cannot be generated by any other integer added to the sum of that other integer's digits. For example, 21 is not a self number, since it can be generated by the sum of 15 and the digits comprising 15, that is, 21 = 15 + 1 + 5. No such sum will generate the integer 209, hence it is a self number. These numbers were first described in 1949 by the Indian mathematician D. R. Kaprekar. students might want to explore self numbers for patterns [The earliest use of Colombian number I can find is by B. Recaman (1974). "Problem E2408". Amer. Math. Monthly 81. Would love to know if there are earlier uses.] 209 is the maximum number of pieces that can be made by cutting an annulus with 19 straight cuts. 209= 105^2 - 104^2 The curve 42x2 - y2 = 209 contains the 'prime points' (3, 13), (5, 29), (7, 43), and (13, 83). *Prime Curios There is an infinity of pairs x,y where x2 - y2 - xy = 209 for x, y integer *Prime Curios 209 is the smallest number with six representations as a sum of three positive squares. These representations are: 209 = 12 + 82 + 122 = 22 + 32 + 142 = 22 + 62 + 132 = 32 + 102 + 102 = 42 + 72 + 122 = 82 + 82 + 92. *Wikipedia 209 is one less than a primorial. Euclid's proof of the infinity of the Primes depends on the fact that n#-1 or n#+1 can not be a factor of any of the primes up to n, hence there must be another. 209 is the smallest n#-1 that is not a prime, but the product of prime factors larger than n=7, (11 x 19) 209 is a palindrome in base 6 (545) (Base six is called senery, After the sequence, binary, ternery, quaternery, quinery, senery, septenery...and then they jump to octal???) There are 209 partitions of 16 into relatively prime parts. The 210th Day of the Year 210 is the last year day that is a Primorial, 210 = #7 = 7*5*3*2. The name "primorial", coined by Harvey Dubner, draws an analogy to primes similar to the way the name "factorial" relates to factors.*Wikipedia Of course that means it is the smallest number that is the product of four distinct primes, and the only such year date. (21, 20, 29) and (35, 12, 37) are the two least primitive Pythagorean triangles with different hypotenuses and the same area (=210). Students are challenged to find another pair of such PPTs There are an infinite number of numbers that appear six or more times in Pascal's Arithmetic Triangle, but only three of them; 1, 120, and 210 are year dates. 7! hours is 210 days. The Combination of ten things taken four at a time is 210. Patrick Honaker asks is there another such where k is not 1 or n-1. *Prime Curios 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 = 210, the sum of eight consecutive primes Three different ways to make a 3x3 magic square with a magic constant of 210, Take the classic 3x3 and multiply each term by 14, 56 126 28 42 70 98 112 14 84 Or with consecutive integers starting at 76 69 74 67 68 70 72 73 66 71 Or maybe with increments of five 65 90 55 60 70 80 85 50 75 The magic is in the middle, all else stems from there. 210 is the 20th Triangular number, the sum of the integers from 1 - 20. 210 in binary is a balanced number, with the same numbers of ones and zeros, and reading from left to right the zeros never outnumber the ones. The sum of the squares of the divisors of 12, is 210. ## Thursday, May 14, 2020 ### Number Facts for Every Year Date, 151-180 The 151st Day of the Year 151 is the 36th prime number, and a Palindromic Prime. Did I ever mention that palindrome is drawn almost directly from an Ancient Greek word that literally means "running back again." First used in English in 1636 in "Camdon's Remains Epitaths". The smallest prime that begins a 3-run of sums of 5 consecutive primes: 151 + 157 + 163 + 167 + 173 = 811; and 811 + 821 + 823 + 827 + 829 = 4111; and 4111 + 4127 + 4129 + 4133 + 4139 = 20639. *Prime Curios... Can you find the smallest 4-run example? 151 is also the mean (and median) of the first five three digit palindromic primes, 101, 131, 151, 181, 191 151 is an undulating palindrome in base 3 (12121) Thanks to Derek Orr, who also pointed out that any day in May (in non-leap year) 5/d is such that 5! + d = year day In 1927 Babe Ruth hit 60 Home Runs, a long lasting record. He hit them in 151 games. And from base to torch, Lady Liberty is 151 ft tall.$151 is the largest prime amount you can make with three distinct US bank notes.

The 152nd Day of the Year: the eighth prime number is 19, and 8 x 19 = 152.... 152 is also the largest known even number that can be expressed as the sum of two primes in exactly four ways. (Students should find all four ways.)

152 is a refactorable number since it is divisible by the total number of divisors (8) it has, and in base 10 it is divisible by the sum of its digits(8), making it a Niven number.

152 is the sum of four consecutive primes, starting with 31.

There are 152 mm tick marks on a six-inch ruler.

152 is the smallest number you can make as the sum of two distinct odd primes cubed.

the digits 152 occur beginning at the digit of e, that is the 152nd prime number (881).

The 153rd Day of the Year 153 The Sum of the aliquot divisors and the product of aliquot divisors are both perfect squares There is a smaller year day with this same property

153 is the fixed point attractor of any multiple of three under the process of summing the cubes of the digits. For more detail and explanation see,"The Cubic Attractiveness of 153" ,

13+53 + 33 = 153. Numbers which are the sum of their own digits raised to the power of the number of digits are called Armstrong numbers. Except for the trivial one digit numbers, it is also the smallest. There are only three other numbers greater than one which are the sum of the cubes of their digits (Go fourth and seek them. hint: this is the only one which is a year date)

And to extend that, the amazing Cliff Pickover shared this:(although the digits are taken in sets)

ALSO, 153 = 1! + 2! + 3! +4! +5!, *Jim Wilder@wilderlab

I had never observed that 153 = 3 x 51, a product that uses all the digits of the number.  HT to INDER J. TANEJA @IJTANEJA There are no other numbers below 1000 that have the same digits as their prime factorization in a simple product (w/o using powers) but there is one lingering just above that number, can you find it?

153 also forms a Ruth-Aaron pair with 154, the product of the distinct Prime factors of each sum to the other.

The 154th Day of the Year:
154 also forms a Ruth-Aaron pair with 153, the product of the distinct Prime factors of each sum to the other.

154 is the smallest number which is a palindrome in base 6, [444]6 ; base 8 ,[242]8; and base 9 ,[181]9 all three. Student's might search for a number that is a palindrome in other simple bases.

154 also has an interesting property with appropriate powers, 1+56+42= 15642. What other day numbers can you find with similar properties?

154 is the twelfth day of the year which is the product of exactly three distinct primes.

154 is the number of ways to partition forty into at most, three parts. (It is also the way to partition 43 into parts of which the greatest part is three).

If You start with 0! = 1, then 154 is the sum of the first six factorials

The largest prime gap below 10,000,000 is 154.

154! + 1 is a prime *Prime Curios

With just 17 cuts, a pancake can be cut up into 154 pieces. This is called the Lazy Caterers sequence.

The 155th Day of the Year
The 155th day of the year; 155 is the sum of the primes between its smallest and largest prime factor. 155 = 5 x 31 and (5+ 7 + 11 + 13 + 17 + 19 + 23 + 29 +31 = 155) *Prime Curios

Fun with primes: 2^2 + 3! + 5! + 7^2 - 11 - 13 = 155.

And from Math Year-Round ‏@MathYearRound 155² +155 ± 1 are twin primes. Students (and teachers) may be surprised how frequently x2+ x ± 1 forms twin primes.

155 is also a pentagonal number, n*(3*n-1)/2, n=0, +- 1, +- 2, +- 3, ..... Euler showed that the pentagonal numbers are the coefficients of the expansion of the infinite polynomial (1-x)(1-x2)(1-x3).... John H. Conway showed that the same series can be found by taking the triangular numbers that are divisible by three, and dividing them.

155 is equal to the sum of the primes from its smallest prime factor, 5, to its largest, 31. There are only three year days of this kind. *HT to Derek Orr

The 156th Day of the Year
The 156th day of the year; 156 is the number of graphs with six vertices. *What's So Special About This Number.

$$( \pi(1)+\pi(5)+\pi(6)) * (p_1 + p_5 + p_6) = 156$$. 156 is the smallest number for which this is true, and the only even number for which it is true. (The symbols $$\pi(n)$$ and $$p_n$$ represent the number of primes less than or equal to n, and the nth prime respectively)

156 is evenly divisible by 12, the sum of its digits. Numbers which are divisible by the sum of their digits are usually called Niven Numbers.
According to an article in the Journal of Recreational Mathematics the origin of the name is as follows. In 1977, Ivan Niven, a famous number theorist presented a talk at a conference in which he mentioned integers which are twice the sum of their digits. Then in an article by Kennedy appearing in 1982, and in honor of Niven, he christened numbers which are divisible by their digital sum “Niven numbers.” One might try to find the smallest strings of consecutive Niven Numbers with more than a single digit. *http://trottermath.net/niven-numbers/
I wonder about the relative order of the classes of numbers which are n times their digit sum for various n.

78 is the 12th Triangular number, which means that twice that, 156, is the number of times a clock that chimed the hours would chime in one day.

156 is a Harshad (Joy-Giver) number, divisible by the sum of its digits.

156/4 = 39 , so 40^2 -38^2 =  156  and 156/12 = 13 so 16^2-10^2 = 156

156 is a repunit in base 5 (1111), and a repdigit in base 25 (66)

The 157th Day of the Year
2157 is the smallest "apocalyptic number," i.e., a number of the form 2n that contains '666'. *Prime Curios (Can you find an apocalyptic number of the form 3n)

157 is prime and it's reverse, 751 is also prime. 157 is also the middle value in a sexy triplet (three primes successively differing by six; 151, 157, 163). 751 is also a sexy prime with 757.

157 is also the largest solution I know for a prime, p, such that $$\frac{p^p-p!}{p}$$ is prime.

!57 is a Repunit in Duodecimal, or base 12 (111); and a palindrome in bases 7 (3137) and 12 (11112).

The number 157 in base ten is equal to $$31_{[52]}$$, but don't worry if you get that backwards, $$52_{[31]}$$ is also equal to 157 in decimal. Can you find other examples of reversible numeral/base that give the same decimal value?

And from Fermat's Library @fermatslibrary In 1993 Don Zagier found the smallest rational right triangle with area 157. He used sophisticated techniques using elliptic curves paired with a lot of computational power. If he could do that, certainly you ought to be able to find the smallest rational right triangle with area of 1.... (OK trick question, ask your teacher to explain)

157 is the largest odd integer that cannot be expressed as the sum of four distinct nonzero squares with greatest common divisor 1 and, The largest odd integer that cannot be expressed as the sum of four distinct nonzero squares with greatest common divisor 1.*Prime Curios

Two to the power 157 is the smallest "apocalyptic number," i.e., a number of the form 2n that contains '666'. *Clifford Pickover

157 is the smallest emirP whose sum of the digits (13) is another emirP. And it's the 37th Prime, another emirP

157 is the largest prime, p, for which $$\frac{p^p+1}{p+1}$$ is prime

157 is the smallest three-digit prime that produces five other primes by changing only its first digit: 257, 457, 557, 757, and 857. *Prime Curios

The 158th Day of the Year
The 158th day of the year; 158 is the smallest number such that sum of the number plus its reverse is a non-palindromic prime: 158 + 851 = 1009 and 1009 is a non-palindromic prime. *Number Gossip (What's the next one?)

Middle school # 158 in Bayside, Queens, New York, is called Marie Curie Middle School.

158 is the sum of the first nine Mersenne prime exponents.

The smallest number such that the sum of the number and its reverse is a prime that is not palindromic, i.e., 158 + 851 = 1009.*Prime Curios

The decimal expansion of 100! (the product of the first 100 natural numbers) has 158 digits.

158 is a number in the Perrin sequence, but lovingly called the "skiponacci" sequence after its resemblance to the Fibonacci sequence. Defined by a(n) = a(n-2) + a(n-3) with a(0) = 3, a(1) = 0, a(2) = 2. The pattern starts 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17,...

The 159th Day of the Year:
A barrel of oil contains 159 Liters.

159 = 3 x 53, and upon concatenating these factors in order we have a peak palindrome, 353, which is itself a prime.*Prime Curios

159 is the sum of 3 consecutive prime numbers: 47 + 53 + 59 and can be written as the difference of two squares in two different ways.

159 is the fifth Woodall number, a number of the form n*2n -1. The numbers were first studied by Allan J. C. Cunningham and H. J. Woodall in 1917, inspired by James Cullen's earlier study of the similarly-defined Cullen numbers n*2n +1. )

Deshouillers (1973) showed that all integers are the sum of at most 159 prime numbers. I'm waiting for someone to tell me the number that takes 159 prime numbers to form???

The 160th Day of the Year
160 is the smallest number which is sum of cubes of 3 distinct primes, the first three. (23+33+53) *Prime Curios (It is also the sum of the first power of the first 11 primes )

160! - 159! + 158! - ... -3! + 2! - 1! is prime. (Quick, guess the approximate size of this number.)

160 is divisible by 4, 8, 16, so $$160 = 41^2 - 39^2 = 22^2 - 18^2 = 14^2 - 6^2$$
And since 160/20 = 8, 160 = = 13^2 - 3^2

160 is also the sum of two non-zero squares (122 + 42) and like all such numbers, you can show that 1602n+1 will also be the sum of two non-zero squares.

160 is a palindrome in base 3 (12221); and in base 6 (424);

In the Collatz problem, starting at 160 takes ten iterations to reach 1, all of them but one is a divided by two step.

The 161st Day of the Year:
Every number greater than 161 is the sum of distinct primes of the form 6n - 1.  *Prime Curios (which numbers less than 161 are also the sum of distinct primes of the form 6n-1? or which are not?)

and for the gamblers out there, There are 161 ways to bet on a roulette wheel.

When 161 is not only a palindrome, when is rotated 180o it gives a palindromic prime, (191) (Such reversible numbers, or words, are called "ambigrams", among other terms.)

Palindrome expression for 161 , 16 + 61 + 7 + 16 + 61

161 is the sum of five consecutive prime numbers: 23 + 29 + 31 + 37 + 41 = 161

The 162nd Day of the Year 162 is the smallest number that can be written as the sum of 4 positive squares in 9 ways.*What's Special About This Number? (Can you find all nine ways?...I should add that five of these use four distinct squares, and the other four have a repeated square..... Can you find a smaller number that can be written as the sum of four squares in eight ways?)

the 12th prime (12 = 1*6*2) ; p12 = 37, and the number of primes less than 162, $$\pi(162)$$ is also 37. There is no smaller number with this property.

Palindrome expressions for 162   3x3x2x3x3 or 9x2x9

162 is the total number of baseball games each team plays during a regular season in Major League Baseball. (But sadly, probably not this year ({2020})

Jim Wilder pointed out that 1621= 162 has a digit sum of nine; and 1622= 26244 has a digit sum of 18; and 1623= 4251528 has a digit sum of 27. And 1624 ???

162 has a sum of divisors 1+2+3+6+9+18+27+54+81=201 which is greater than 162. Such numbers have been called abundant since the Ancient Greeks.

A 3x3 magic square with a magic constant for each row and column of 162

53    58   51
52   54    56
57   50    55

Imagine you have seven distinctly colored balls, and three numbered tubs to put them in, but none can be in a tub by itself. There are 162 different ways to distribute the balls. (If students struggle with this large a challenge, they can try to find all eleven ways to put five colored balls in just two tubs, again with no solitary balls.

A T Vandermonde should be remembered for the wonderfully useful approach he had for generalizations on the factorial, and in my mind created the most useful notation ever (and, he seems to have been the first to think of 0!=1) His notation included a method for skipping numbers, so that [p/3]n would indicate p(p-3)(p-6)... (p-3(n-1)); and in his notation 162 = [9/3]3 or 9*6*3. Now that's a notation worth having an exclamation point. Today this is called a triple factorial, but it doesn't, to my knowledge, have a way to stop along the way, like 16*13*10.

The 163rd Day of the Year
The 163rd day of the year; \$e^{\pi*\sqrt{163}} \$ is an integer. Ok, not quite, In the April 1975 issue of Scientific American, Martin Gardner wrote (jokingly) that Ramanujan's constant (e^(π*sqrt(163))) is an integer. The name "Ramanujan's constant" was actually coined by Simon Plouffe and derives from the above April Fool's joke played by Gardner. The French mathematician Charles Hermite (1822-1901) observed this property of 163 long before Ramanujan's work on these so-called "almost integers." Actually equals 262537412640768743.99999999999925..

.     and      *WIkipedia

Colin Beveridge ‏@icecolbeveridge pointed out that $$(2+\sqrt{3})^{163}$$ is also very, very close to an integer. (but it is very large,greater than 1093 , and was not, to my knowledge, ever the source of an April fools joke.)

163 is conjectured to be the largest prime that can be represented uniquely as the sum of three squares \$163 = 1^2 + 9^2 + 9^2 \$.

Most students know that the real numbers can be uniquely factored. Some other fields can be uniquely factored as well, for instance, the complex field a+bi where i represents the square root of -1 is such a field. In 1801, Gauss conjectured that there were only nine integers k such that $$a + b\sqrt{-k}$$ is a uniquely factorable field.; The largest of these integers is 163.; Today they are called Heegner numbers after a proof by Kurt Heegner in 1952.

163 is as easy as 1+2*3^4.

163 is a strictly non-palindromic number, since it is not palindromic in any base between base 2 and base 161.

163 figures in an approximation of π, in which \$\pi \approx {2^{9} \over 163}\approx 3.1411}\$. *Wik

163 is the 38th prime number

The 164th Day of the Year
With the ordered digits of 164 we can form 3 2-digits numbers. Those 3 numbers ± 3 are all prime (16 + 3 = 19, 16 - 3 = 13, 14 + 3 = 17, 14 - 3 = 11, 64 + 3 = 67, 64 - 3 = 61). *Prime Curios

In base 10, 164 is the smallest number that can be expressed as a concatenation of two squares in two different ways: as 1 + 64 or 16 + 4

There are 164 ways to place 5 non-attacking queens on a 5 by 8 board. */derektionary.webs.com/april-june

164 can be expressed as the concatenation of two squares in two different ways, 1, 64 or 16,4. The smallest number for which that is true. Can you think of the next?

Because 164/4 = 41, 164=42^2 - 40^2.

164 = 10^2 +8^2

A scrabble board has 225 squares on the board, many are special squares with double letter or double word notation, but 164 have nothing.

164 is CLXIV in Roman Numerals, using every symbol 100 or below once each.

164 is a palindrome in base 3(20002), and in base 9(202),

The 165th Day of the Year
The 165th day of the year; 165 is a tetrahedral number, and the sum of the first nine triangular numbers. The tetrahedral numbers are found on the fourth diagonal of Pascal's Arithmetic Triangle, and given by the combinations of (n+2 Choose 3) or Tetn = $$\frac {(n)(n+1)(n+2)}{6}$$ also easy to remember this is the nth triangular number times (n+2)/3.

165 is a sphenic number, the product of three distinct primes.

Numbers greater than 35 that end in five are the difference of two squares five apart, 165-25 = 140 so  19^2-14^2 = 165  also, all numbers in the sequence of f(n)= 6n+9 are (n+3)^2 - n^2. so 169 = 209^2 - 26^2

165 is also the sum of the squares of the first five odd numbers.

165 is the sum of the divisors of the first fourteen integers.

165 is sort of a prime average(or an average of primes) The two nearest primes are 163 and 167, with 165 as their average; The next two nearest are 157 and 173, yeah, 165 is their average; The next two out are 179 and 151, yes again, average is 165; then 149 and 181, yep!.... 139 and 191, yep!.... 137 and 193...Oh Yeah!... 131 and 197... awww heck, but if you slipped 199 in in place of 197, you'd get one more.

A 5x5 magic square with arithmetic sequence  and magic constant of 165

41  55   9  23  37
53  17  21  35  39
15  19  33  47  51
27  31  45  49  13
29  43  57  11  25

165 has three prime factors, called a sphenic number. Its reversal, 561 also has three prime factors.

165 is a tetrahedral number, the sum of the triangular numbers from 1 to 45.

165 is a palindrome in base 2(10100101) and bases 14 (BB14), 32 (5532) and 54 (3354).

The 166th Day of the Year;
the reverse (661) of 166 is a prime. If you rotate it 180o (991) it is also prime. The same is true if you put zeros between each digit (10606).  *Prime Curios

166!-1 is a factorial minus one prime.  (For which n is N! -1 or n! + 1 a prime? hint: there are three more year days for which n! +1 or n! -1 is prime

166, like 164, uses all the Roman digits from 100 down, once each. A difference is that 166 uses them in order of their size, CLXVI.

166 is a palindrome in base 6 (434) and base 11(141)

The 167th Day of the Year,
167 is the smallest of a sextet of numbers related to the well known Ramanujan taxicab number. Ramanujan had commented that 1729, the number of Hardy’s taxi, was the smallest number that can be expressed as the sum Of two positive cubes in two ways. But what about three ways? Unfortunately, I don’t know who found this (help?). 167^3 + 436^3 = 228^3 + 423^3= 255^3 + 414^3 =87539319.

So here's a challenge, Wieferich proved that 167 is the only prime requiring exactly eight cubes to express it. Can you find the eight? *Prime Curios There are 16 year dates that can not be expressed with 17 non-negative cubes.

167^4 = 777796321, the smallest number whose fourth power begins with four identical digits, *Prime Curios
 *MAA Found Math

167 is the 39th prime.
It's reversal, 761 is also prime, so 167 is an emirP, and it's the smallest of them whose emirP index, it's the 13th emirp) is also an emirP
167 (prime) and 169 (square) meet, at least in New York City

Remember, LaGrange said that every natural number can be expressed with four squares. So find them for 167.

The reciprocal of 167 is a repeating decimal with a digit cycle of 166 digits.

The 168th Day of the Year
there are 168 prime numbers less than 1000. *Prime Curios

168 is the product of the first two perfect numbers. *jim wilder ‏@wilderlab

$$2^{168} = 374144419156711147060143317175368453031918731001856$$ lacks the digit 2; no larger 2n exists for $$n \lt 10^{399}$$ that is not pandigital.

168/4 = 42 so 168= 43^2-41^2; 168 /8 = 21 so 168 = 23^2 - 19^2, and 168/12 = 14 so 168 = 17^2 - 11^2
There are 168 hours in a week.

168 is also the number of moves that it takes a dozen frogs to swap places with a dozen toads on a strip of 2(12) + 1=25 squares (or positions, or lily pads) where a move is a single slide or jump. This activity dates back to the 19th century, and the incredible recreational mathematician, Edouard Lucas *OEIS. Prof. Singmasters Chronology of Recreational Mathematics suggests that this was first introduced in the American Agriculturalist in 1867, and I have an image of the puzzle below. The fact that they call it, "Spanish Puzzle" suggests it has an older antecedent. (anyone know more?)

The 169th Day of the Year

The 169th day of the year; 169 is the smallest square which is prime when rotated 180o (691)  What is the next one?

And from Jim Wilder, 169 is the reverse of 961. The same is true of their square roots... √169=13 and √961=31 or stated another way, 169 = 132 and in reverse order 312 = 961

An interesting loop sequence within Pi. If you search for 169, it appears at position 40. If you then search for 40, it appears at position 70. Search for 70, ... 96, 180, 3664, 24717, 15492, 84198, 65489, 3725, 16974, 41702, 3788, 5757, 1958, 14609, 62892, 44745, 9385, 169, *Pi Search page

169 is the only year day which is both the difference of consecutive cubes, and a square: $$8^3-7^3 =169=13^2$$

The first successful dissection of a square into smaller squares was of a square with 169 units on a side. 1907-1914 S. Loyd published The Patch Quilt Puzzle. A square quilt made of 169 square patches of the same size is to be divided into the smallest number of square pieces by cutting along lattice lines. The answer, which is unique, is composed of 11 squares with sides 1,1,2,2,2,3,3,4,6,6,7 within a square of 13. It is neither perfect nor simple. Gardner states that this problem first appeared in 1907 in a puzzle magazine edited by Sam Loyd. David Singmaster lists it as first appearing in 1914 in Cyclopedia by Loyd but credits Loyd with publishing Our Puzzle Magazine in 1907 - 08. This puzzle also appeared in a publication by Henry Dudeney as Mrs Perkins Quilt. Problem 173 in Amusements in Mathematics. 1917

169 = (2^7 + 7^2) - (7 + 1) and is the smallest perfect square of the form (2p + p2) - (p + 1).*Prme Curios

169 is the sum of seven consecutive primes, 13 + 17 + 19 + 23 + 29 + 31 + 37

169 is the last square in the Pell sequence

169 is a palindrome in base 12(121) "A gross plus two dozen and one more."

The 170th Day of the Year
The 170th day of the year; the start of a record-breaking run of consecutive integers (170-176) with an odd number of prime factors.

170 is the smallest number that can be written as the sum of the squares of 2 distinct primes, where each of these primes is the square of a prime added to another prime (170 = (22 + 3)2 + (32 + 2)2). *Prime Curios

170 is the largest integer for which its factorial can be stored in double-precision floating-point format. This is probably why it is also the largest factorial that Google's built-in calculator will calculate, returning the answer as 170! = 7.25741562 × 10306. (For 171! it returns "infinity".)

170 is the smallest number n for which phi(n)(the number of integers relatively prime to 170=64=82) and sigma(n) (the sum of the divisors of 170=324=182) are both square.

Just as a curiosity, the $$\sigma_0(170)$$, sometimes called the divisor function is 8, that is, there are eight numbers that divided evenly into 170, 1, 2, 5, 10, 17, 34, 85, and 170, and the number of totitives (coprime values) is 8^2.  Some people also use $$\sigma_2(n)$$ for the sum of the squares of the divisors of n.  For 17, that's 37,700 , and yes, they do similar things for any power that amuses them.

The 171st Day of the Year

$$10^{171 } - 171$$is a prime number with 168 nines followed by 829

Google calculator gives 171! = infinity. (close enough in many cases)
171 - 9 = 162, and 162/6 = 27, so 30^2 - 27^2 = 171

Das Ambigramm added that 163 = 3 x 7 x 11 x 11 x 7 x 3, but also 5 x 5 + 11 x 11 + 5 x 5

171 is the last year-day that is both a triangular number and a palindrome. *Ben Vitale

171 is the 18th triangular number, the sum of the integers from 1 to 18.

171 is a repdigit in base 7 (333)

171 is a Harshad (joy-giver) number, divisible by the sum of it's digits.

The 172nd Day of the Year
seventeen 2's followed by two 17's is prime.*Prime Curios 222222222222222221717 is prime

$$172 = \pi(1+7+2) * p_{(1*7*2)}$$. It is the only known number (up to 10^8) with this property. pi(n) is the number of primes less than or equal to n, and pn is the nth prime.

172/4 = 43, so 44^2 - 42^2  = 172

172 is the sum of Euler's Totient function (the number of smaller numbers for each n, which are coprime to n) over the first 23 integers

172 is the number of pieces a circle can be divided into with 18 straight cuts. It is sometimes called the Lazy Caterer's sequence, and is given by the relation $$p = \frac{n^2+n+2}{2}$$
Since I haven't mentioned this anywhere else yet, these numbers appear in Floyd's Triangle, a programing exercise for beginning programmers which has the Lazy Caterer sequence going veritcally down the altitude of a triangle of numbers, and the triangular numbers on the hypotenuse
1
2, 3
4, 5, 6
7, 8, 9, 10
11.....

172 is a repdigit in base 6(444), and also in base 42 (44)

The 173drd Day of the year
the only prime whose sum of cubed digits equals its reversal: 13 + 73 + 33 = 371. *Prime Curioos

The smallest prime inconsummate number, i.e., no number is 173 times the sum of its digits. (The term inconsummate number was created by John Conway from the Latin for unfinished. [when?])

173 is the largest known prime whose square (29929) and cube (5177717) consist of totally different digits.

173 is a Sophie Germani Prime since 2*173+1 = 347 is also prime. Sophie Germain primes are named after French mathematician Sophie Germain, who used them in her investigations of Fermat's Last Theorem. Sophie Germain primes and safe primes have applications in public key cryptography and primality testing. It has been conjectured that there are infinitely many Sophie Germain primes, but this remains unproven.

173 is the sum of the squares of two Fibonacci numbers. (Which two?) and the difference of two squares, 173 = 87^2 - 86^2

173 = 1 + 2^2 + 2^3 + 2^5 + 2^7. The exponents are consecutive primes.*Prime Curios

If I understood what I read, 173 is the largest known prime for which its square, 29929, and its cube, 5177717 have no digits in common,

And hey, 173 is the prime first three digits of the square root of three (also prime, but you knew that).

173 is the sum of three consecutive primes, 53 + 59 + 61 = 173 (Wondering what percent of the time three consecutive primes add up to a prime? Seems fairly common with low digits.)
173 is a palindrome in base three(20102) and base 9 (212)

The 174th Day of the Year
there are 174 twin prime pairs among the first 1000 integers.

174 = 72 + 53 (using only the first four primes) and is also the sum of four consecutive squares.

174 is the smallest number that begins a string of four numbers so that none of them is a palindrome in any base, b, $$2 \leq b \leq 10$$

174 is a sphenic (wedge) number, the product of three distinct prime factors, 174 = 2*3*29.
174 is called an "integer perfect number" because its divisors can be partitioned into two sets with equal sums.

174 is the smallest number that can be written as the sum of four distinct squares in six different ways,
174 = 1^2 + 2^2 + 5^2 + 12^2
= 1^2 + 3^2 + 8^2 + 10^2
= 1^2 + 4^2 + 6^2 + 11^2
= 2^2 + 5^2 + 8^2 + 9^2
= 3^2 + 4^2 + 7^2 + 10^2
= 5^2 + 6^2 + 7^2 + 8^2
*Srinivasa Raghava K
The 175th day of the year;
175 is the smallest number n greater than 1 such that n^6 $$\pm 6$$ are both prime. *Prime Curios & Derek Orr

175 - 25 = 150 and 150 /10 = 15 so 20^2-15^2 = 175

175 is the number of partitions of 35 into prime parts.

From Jim Wilder ‏@wilderlab : $$175 = 1^1 + 7^2 + 5^3$$

A normal Magic square of order 7 has a "magic constant" of 175 for the sum of each row, column or diagonal.  The one below comes from the "Geeks For Geeks" web site, but this particular Geek wishes they had rotated it one-quarter turn clockwise so that the smallest number is in the center of the bottom row.
And if you want a unique way to create any normal magic square (and with a little imagination, lots of other odd order magic squares) for a nice way to create the one above , but rotated  https://pballew.blogspot.com/2018/06/suprise-unique-approach-for-odd-order.html

The 176th Day of the Year
176 and its reversal 671 are both divisible by 11. ( Students should confirm that the reverse of any number that is divisible by 11 will also be divisible by 11.)

176 is a happy number, repeatedly iterating the sum of the squares of the digits will lead to 1, 12 + 72 + 62= 86, 82 + 62 = 100 and 12 + 02 + 02 = 1

The number 15 can be partitioned in 176 ways. For younger students, imagine all the different ways of making fifteen cents with US coins, 1 cent, 5 cnets, and 10 cents.... now imagine there were also coins worth 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, and 15 cents. There would be 176 different collections of coins that would total exactly 15 cents.

The divisors of 176 are 1, 2, 4, 8, 11, 16, 22, 44, 88, and 176. Take away 1, 176, 8 (the sum of the four 2's that are prime factors, and 11, (the other prime factor) now add up what's left, 2+4+16+22+44+88= 176, yeah, making magic happen here.

176 is a Self number, it can't be written by any other number plus the sum of its digits. 21 for instance, is not a self number because 15+1+5 = 21.

8*20 + 16 = 176 so 176 = 24^2 - 20 ^2  (try this on 192 also ) and it is divisible by 16 with quotient 11, so 15^2 - 7^2 = 176.

176 is also a happy number, if you square each of its digits, and add them, then do the same to the result, eventually you will have the happy ending of getting 1, which would simply repeat itself forever, and knowing this, you will stop... please stop.... STOP!

176 is also a cake number, the number of ways of slicing a cube with 10 planes to get the greatest number of cake pieces. A three dimensional analogy to the lazy caterer's number in two space.

The 177th Day of the Year
there are 177 graphs with seven edges.  *What's So Special About This Number.  (only 79 of these are connected graphs)

• 177 is the smallest magic constant for a 3 x 3 prime magic square
$\begin{bmatrix} 17 & 89 & 71 \\ 113 & 59 & 5 \\ 47 & 29 & 101 \end{bmatrix}.$

177 is the sum of the primes from 2 to 47 taking every other one, 2+5+11+.... +41+47, and 177 is the 15th or 1+7+7th prime.  *Prime Curios

177 is a deficient number, the sum of its aliquot divisors is 59+3+1 = 63, far less than 177.  Its deficiancy is 114= 177-63.  All odd numbers up to 945 are deficient, the second smallest deficient odd number is 1575  All of the known abundant odds are divisible by 5.

177 is also a Leyland number,  expressible as a^b + b^a. both greater than one . using 2 and 7 in this case.  . The numbers are named for British mathematician Paul Layland from Oxford University.  There are only nine year days that are Leyland numbers.  Only one of those nine is prime.

The 178th Day of the Year
178 = 2 x 89. Note that 2 and 89 are the smallest and the largest Mersenne prime exponents under 100. *Prime Curios

178 is a palindrome in base 6,$$[454]_6$$ and in base 8 $$[262]_8$$

Strangely enough, 178 and 196 are related... In fact, 178 has a square with the same digits as 196: 1782 = 31,684 1962 = 38,416 178 has also a cube with the same digits as 196: 1783 = 5,639,752 1963 = 7,529,536 *Zoo of Numbers

178 = 13^2 + 3^2

178 is a palindrome in base 6 (454), base 7 (343), and base 8 (262)

178 is a semi-prime, the product of 2 and 89, which are the smallest, and largest Mersenned prime exponents under 100.

178 is a digitally balanced number, it binary expression has an equal number of zeros and ones, 10110010, and they are balanced so that the first and last, 2nd and 2nd last, etc always have a one and a zero.

The 1179th Day of the Year
179 is a prime whose square, 32041, has one each of the digits from 0 to 4.

179 is a "Knockout Prime" of the form K(3,2) since 17, 19, and 79 are all prime.

179 is an emirp, a prime whose reversal, 971 is also prime, and the combination sum and product 179 * 971 + 179 + 971=174959 is also an emirp.

1793 has all odd digits, 5735339. *Derek Orr

179 = (17 * 9) + (17 + 9)

A winning solution to the 15-hole triangular peg solitaire game is: (4,1), (6,4), (15,6), (3,10), (13,6), (11,13), (14,12), (12,5), (10,3), (7,2), (1,4), (4,6), (6,1). The term (x,y) means move the peg in hole x to y. Not only does this solution leave the final peg in the original empty hole, but the sum of the peg holes in the solution is prime. But not just any prime, it is 179.

Between the beginning and the 179th digit of π, an equal number of five different decimal digits occur (there are 18 each of the digits 0, 3, 4, 5, and 9). Mike Keith conjectures this to be the last digit of π for which this happens (there are no others up to 10^9 digits). *Prime Curios

1/179 has a repeating patter of 178 digits, called a full repetend prikme.

179 is a strictly non-palindromic number. It is not a palindromic number in any base.*Wikipedia

The 180th Day of the Year
180 can be formed with the only the first two primes... 180 = 22 x 32 x (2+3) *Prime Curios

180 is the sum of two square numbers: $$12^2 + 6^2$$. It can also be expressed as either the sum of six consecutive primes: 19 + 23 + 29 + 31 + 37 + 41, or the sum of eight consecutive primes: 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37.
As differences of two squares 180= 46^2 - 44^2

180 is a Harshad (Joy-Giver in Sanskrit) as it is divisible by the sum of its digits.

Beautiful trigonometry, arctan1 + arctan2 + arctan3 = 180o

180=2^2*3^2 *(2+3)

180 has more divisors than any smaller number. It is also called a refactorable number because it is divisible by the number of divisors it has, 18.

Pi radians is equivalent to 180o

As was 178, 180 is a digitally balanced number, its eight binary digits contain four ones and four zeros, 10110100, and like 178, they match up into two sets of balanced zero-one pairs, the first four digits, 1011, aligning perfectly with their digital opposite in the last four 0100.