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 also the hypotenuse of a near-isosceles right triangle with the longer side of 180, and the shorter side of 19.

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)

A good approximation to the number of seconds in one year from Srinavasa Raghava, $$\pi ^{181/12}$$

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

Pronic numbers are the product of two consecutive integers. They are always twice a triangular number.  182 is the smallest pronic (13 x 14) number (not ending in zero) whose reversal is a prime. Most Pronic numbers are abundant, and only three year days are deficient, 182, 110, and 2.

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)

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

186 is the median between consecutive primes 181 and 191.

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.

187 = 11 x 17 is a semiprime and a reversible semiprime 781 = 11x71
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 , 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)

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..

188 can be rotated 180 degrees to get another number, 881.  Numerals like 181 which stay the same when rotated are called strobogrammatic numerals

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.

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

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).

189 is the sum of consecutive cubes, 4^3 + 5^3 = 189.

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 sum of five consecutive squares, 190 = 4^2 + 5^2 + 6^2 + 7^2 + 8^2

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 n^2 triangular numbers 0, 1, 3, 6, 10, . . . n^2(n^2 – 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. Such numbers where 2p+1 is also prime are called Sophie Germain primes. She studied the numbers in relation to Fermat's Last Theorem and proved it was true for the numbers of this type.

It is also a palindrome when rotared 180 degrees into 161.  Such numbers are called ambinumerals.  If the rotation produces the same number, think 181, then it ts called a strobogrammatic numeral.

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 also 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. This is the last prime quadruplet that are year days.
The prime quadruplet {11, 13, 17, 19} is alleged to appear on the Ishango bone although this is disputed.

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. 192 + 384 = 576.  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 square 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

Your students probably know about perfect numbers, but do they know about pseudoperfect numbers?  192 is one because you can add a subset of its divisors and get 192.

192 is called a practical number, because you can make a subset of its divisors to sum to any number less than 192. All perfect numbers are practical, as are all Harmonic divisor numbers.
$$n*2^{n-1}$$ gives the number of edges (segments) in a n-dimensional cube, and in the 6th dimension, called a hexeract, there are 192 edges, 6*25

192 = 2^6 x 3.  With so many factors of two, it has many expressions as the difference of two squares, 192 = 49^2 - 47^2 = 26^2 - 22^2 = 19^2 - 13^3 = 16^2 - 8^2 = 14^2 - 2^2

The 193rd day of the Year
193 = 12^2 + 7^2, like all primes of 4k+1 form, There can only be expressed as a sum of two squares in one way.

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.

193 is the second prime in a prime quadruplet.  191, 193, 197, 199

(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=5^2 + 13^2
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 = 7^2 + 8^2 + 9^2 , there are 12 year days which are the sum of three consecutive squares, 8 are smaller than 194, three are larger.

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 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
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 base 2, 4, 8, and 14.

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!

1956 =54980371265625 is the smallest sixth power that begins with 10 distinct digits.

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 + 11^2  It is only  the third integer that is the sum of the squares of three consecutive primes.*Prime Curios

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

"Betrothed Pairs" are similar to amicable numbers, all the non trivial divisors (don't count 1 or n)  sum to the other member, 195 and 140 are such a pair. There is only one other betrothed pair which are year days.

(1²+2²+3²+...+337²)/337=195²

a Heronian triangle is a triangle that has side lengths and area that are all integers. There is an almost-equilateral, scalene Heronian  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.,

If you take an integer, almost any integer it seems, and add the number to its reverse, then continue this process with each succeeding sum until you come to a palindromic number, you almost certainly will.  38 takes only one addition, 38 + 83 = 121.  139 takes two, 139 + 931 = 1070 and 1070 + 0701 = 1771.  But take a bite out of 196, and you may have to give up before you find it.  Computers have gone through thousands of iterations and, to my knowledge, never found a palindrome.
Any suggestions for why this one number out of all the integers, might not have a palindromic terminal?

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!)(And now I know, it is 41)

199 is the sum of the digits of all the three-digit palindromic primes.  *Prime Curios

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? What is the smallest odd unprimeable number? )

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 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 fourteen Smith Numbers that are year days, and four are palindromes, one trivial.  202 is a smith number because 202 = 2 x 101 and 2+2 = 2 + 1 + 0 + 1   A composite number with the sum of its digits equal to the sum of the digits of it's prime factors
Numbers where each prime factor is used only once, are called hoax numbers, see 364.  Smith numbers were named by Albert Wilansky of Lehigh University, as he noticed the property in the phone number (493-7775) of his brother-in-law Harold Smith:
4937775 = 31 52 658371

while

4 + 9 + 3 + 7 + 7 + 7 + 5 = 3 · 1 + 5 · 2 + (6 + 5 + 8 + 3 + 7) · 1 = 42

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

205  is the median between two consecutive primes, 199 and 211

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), and in bases 9, 11, 13, and 16.

205 is the smallest un-primeable odd composite  .  It can not be turned into a prime by changing only one of its digits.  (Language fact: A prime which cannot be turned into another prime by changing a single digit is called weakly prime. These are all pretty big, the smallest being 294001.)

205 is a semi-prime, 5 x 41.  It is also a reversible semi-prime, or emirpimes, since it's reversal, 502 = 2 x 251, is also a semi-prime

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 is an unprimeable number, changing any digit to something else will not make a prime.

208 = 2^4 x 13  and if you play the four-fours game, 208 = 4^4 - 4! - 4!

208 can be written as the sum of two squares in only one way, 12^2 + 8^2

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 prime 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 a Harshad (Joy-giver) number (another creation of Kaprekar) , divisible by the sum of its digits. The smallest with digit sum of 11. H/T John Golden

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 42x^2 - y^2 = 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 = 1^2 + 8^2 + 12^2 = 2^2 + 3^2 + 14^2 = 2^2 + 6^2 + 13^2 = 3^2 + 10^2 + 10^2 = 4^2 + 7^2 + 12^2 = 8*2 + 8^2 + 9^2. *Wikipedia

209 is one less than a primorial, 7# - 1 . 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.

As far as I know, there are only two three digit numbers so that abc^2 = uvwxyz and uvw+xyz = abc.  These are called three digit Kaprekar numbers.  209 is involved in each case.  The two numbers are 297^2 = 88209, with 88 + 209 = 297 and 703^2 = 494209 where 494 + 209 = 703

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.

210 is a Harshad (joy-giver) number, divisible by the sum of its digits.  In fact, it is a multiple Harshad number since 210/3 = 70, which is also a Harshad number.

(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. It is also C(21,2)

13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 = 210, the sum of eight consecutive primes

210 is the 20th Triangular number, the sum of the integers from 1 - 20.
210 is the last year date which is both a triangular number and the product of consecutive numbers, 14 x 15.  It is also the last to be the product of three consecutive numbers,  5 x 6 X 7.

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 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.