## 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) (Students who have explored Happy Numbers may enjoy exploring the number chains formed by the sum of the cubes of the digits. Other numbers, like 352 pass through several iterations before closing the loop back to themselves, and of course, some numbers never make it back home. 352 --> 160 --> 217 -->352 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 Not only is 153 the sum of the cube of its digits, and the sum of five consecutive factorials, it is the sum of the first 17 positive integers, 1 + 2 + .... + 16 + 17 = 153. That makes 153 the 17th triangular number, and its reversal, 351, is also a triangular number, the 26th. 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. At one time, a new perfect number of 155 digits was announced. On March 27,1936 The Associated Press released a story that a new 155 digit perfect number had been found by Dr. S. I. Krieger of Chicago. The number was $$2^{256}(2^{257} - 1)$$ by proving the $$2^{257} -1$$ was prime. This was shocking since D. H. Lehmer and M. Kraitcik had announced that the number was composite in 1922. Unfortunately, their method did not include giving a factor of the number. The perfection of the number was doubted by most mathematicians, but the actual factoring to prove it was composite didn't happen until 1952 when the SWAC confirmed it was composite by finding a proper divisor. *Beiler, Recreations in the Theory of Numbers. According to current lists, the closest number of digits for a perfect number are an 77 digit number found by Edouard Lucas in 1876, and a 314 digit number found by R M Robinson in 1952. 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 28 x 157 = 4396 uses all nine non-zero digits. 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??? 48 x 159 = 5346, uses all nine non-zero digits 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); 160 is the largest year day (and second largest known) for which the alternating factorial sequence is prime: 160!- 159! + 158! - 157! .... + 2! - 1!. The alternating factorial 5! - 4! + 3! - 2! + 1! = 121. The alternating factorial sequence is prime for n= 3 through 8 (5, 19, 101, 619, 4421, 35899). In spite of this run of consecutive primes, John D Cook checked and found only 15 n values for which the alternating factorial starting with n is prime. 14 are year days, the largest being 160. The one non-year day it turns out uses the same digits as 160, 601. In the Collatz problem, starting at 160 takes ten iterations to reach 1, all of them but one is a divided by two step. 160 is the sum of the cubes of the first three primes. $$2^3 + 3^3 + 5^3 = 160$$ 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?) [spoiler, the nine ways are shown at the bottom of this entry] 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 $$\pi \approx \frac{2^9}{163} \approx 3.1411$$. *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 $$\frac{164}{4} = 41, 164=42^2 - 40^2$$.

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 $$20002_3$$, and in base $$202_9$$,

T(164) (the 164th triangular number) is the hypotenuse of a triangle with all triangular numbers for its side lengths.  The legs of the triangle are T(132) and T (143).  $$8778^2 + 10296^2 = 13530^2$$

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?)

168 and 249 have an interesting relationship, the sum of their digits are equal, and the sum of the squares of their digits are equal. $$1 + 6 + 8 = 2 + 4 + 9 = 15$$ and $$1^2 + 6^2 + 8^2 = 101 = 2^2 + 4^2 + 9^2$$

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 only multidigit year-day that is both a triangular number and a palindrome; and it is one of only eleven triangular year days that is divisible by the sum of its digits.

It was Gauss who discovered that all natural numbers are the sum of at most three triangular numbers, a discovery he announced with pride on July 10, 1796 when he wrote in his diary, EYPHKA (Eureka), num = $$\bigtriangleup + \bigtriangleup = \bigtriangleup$$
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 173rd Day of the year
the only prime whose sum of cubed digits equals its reversal: 13 + 73 + 33 = 371. *Prime Curioos

137 is the sum of the squares of the first seven digits of pi, $$3^2+ 1^2 + 4^2 + 1^2 + 5^2 + 9^2 + 2^2 = 137.$$  *Prime Curios (There is no smaller number of digits of pi for which this is true.) If you add the square of the next digit (6^2) you get another prime which is a permutation of the digits of this one, 173.  These two are the only prime year days which are the sum of the squares of the first n digits of Pi.

Another permutation of 173 is 371, and 173 is the hexdecimal expression of 371 in decimal.

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

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 : $$175 = 1^1 + 7^2 + 5^3$$ There is one more three digit year date which has this same relation.  Find it.

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.