## Wednesday, November 18, 2020

### Number Facts of the Year - 331 - 366

The 331st Day of the Year.
331 is a cuban prime of the first kind, [(y+1)^3 - y^3]  (In this case, y = 10).  Like all cuban primes of the first kind it is a centered hexagonal number.   It is also a centered pentagonal number.

31 is prime, and 331 is prime, and 3331 is prime and likewise for all the way up to 33333331. The next (with eight threes) is not prime, but if you permute the last two digits, you get 333333313, which is prime. Next string of threes followed by a single one is 17 threes and a one. strings of threes ending in 313, 311, and 323 seem likely to be prime.
A nice symmetric pic from Jim Wilder@wilderlab:

John Golden @mathhombre used this idea for one of his Talking Numbers Comics.  Enjoy

331 is the sum of five consecutive primes, 59, 61, 67, 71, 73.
It is also the sum of the first 15 semi-primes *Derek Orr

331 is the 67th prime and the sum of its digits is 7, a prime number with a prime order and a prime sum of its digits, and in the old days (before 1900) all its digits would have been considered prime.

331 is a Happy number, it goes to one under the repeated iteration of the sum of the squares of its digits.  3^2 + 3^2 + 1 = 19, 1^2 + 9^2 = 82, 8^2 + 2^2 = 20....... eventually to one.

Alexandria, Egypt (birthplace of Euclid) was founded in 331 BC by Alexander the Great. *Prime Curios

331 = 166^2  - 165^2

The 332nd Day of the Year
332= 2^2 x 83, and is the difference of two squares, 84^2 - 82^2, and the sum of three squares in more than one way, 2^2 + 2^2 + 18^2 = 6^2 + 10^2 + 14^2.

The sum of the first 332 primes is prime. *Derek Orr

332 is the number of ways to partition 47 into non-zero triangular numbers.  (36 + 10 + 1 would be one such way)

The sum of the proper divisors of 332 is 256 = 2^8, *Derek Orr

In base 3, 332 is a triple double digit number, 110022

332 and 333 have the same number of divisors, 6.

As numbers get larger and larger, it would seem that there would be fewer and fewer primes in each century of them, such as from 100 to 199.  But there seem to be a large number of centuries with only six primes.  There are only five year days such that between 100*n and 100*n+99 there are exactly six primes. 332 is one of them.  (there are exactly six primes between 33200 and 33299)

The 333rd Day of the Year
333=3^2 x 37, It is a Joy-Giver number, divisible by the sum of its digits.

333 is  palindrome, and the sum of its proper divisors, 161, is also a palindrome.  *Derek Orr (and Derek, the sum of all its divisors, 494, is also a palindrome.  )

333 is a Polignac number.  It can not be formed by the sum of a power of two and a prime. (See Day 127 or 337 for Polignac's conjecture about this).

333 in base eight is also a palindrome, 515.

333 = 101 + 11 + 2 + 3 + 41 + 5 + 61 + 7 + 83 + 19, where each term is the smallest prime containing 0, 1, 2, ..., and 9, respectively. *Prime Curios

333 = 3^2 + 18^2 = 167^2 - 166^2= 57^2 - 54^2 = 23^2 - 14^2

333 is also the sum of three squares in different ways, 4^2 + 11^2 + 14^2 = 8^2+ 10^2 + 13^2

333 is expressible as n(4*n+1) and also as (4n+1)(4m+1)

There are 333 possible hexagonal polyominoes with seven cells.

Not only is 3^2 + 4^2 =5^2 and,
33^2 + 44^2 = 55^2  but
333^2 + 444^2 = 555^2

33² + 544² = 545²

333² + 55444² = 55445²

3333² + 5554444² = 5554445²

33333² + 555544444² = 555544445²

333333² + 55555444444² = 55555444445²

3333333² + 5555554444444² = 5555554444445²

The 334th Day of the Year
334 is an even semi-prime, 2 x 167, and together with 335 they form a semi-prime pair. (There will be one more day pair this year that is a semi-prime pair, can you find it?) There are only three more even semiprimes in the year.

D. R. Kaprekar  created the name "self number" for numbers that can not be made up as the sum of any number n and the sum of its digits. They are also called Colombian numbers or Devlali numbers. 1, 3, 5, 7, 9, 20, 31, are some of the smaller self-numbers, and of course, 334 is a self number.

334 in base three looks more like a binary number, its 110101. Students might explore other numbers in base three (or four or five) that look binary (ie only made up of zeros and ones).

334 = 2 + 3 x 5 + 7 x 11 + 13 x 17 +19  *Derek Orr

334 uses the same digits in base 13 and 14, 1C9 and 19C *Derek Orr

1^2 + 3^2 + 18^2 = 3^2 + 6^2 + 17^2 = 3^2 + 10^2 + 15^2 = 334

334^4 + 1 is prime, it is the 48th such number of the year. A year has 51 such numbers .

The 335th Day of the Year
335 = 5 × 67,

335 is the sum of four, but no fewer, squares.  It is the 54th year day for which this is true.

335 is the sum of all the digits from 1 to 38.

335^3 is an eight digit number that has all odd digits . *Derek Orr

2335 is the smallest power of two which equals the sum of four consecutive primes. *Prime Curios This seems astounding to me, that such a huge number would be the first.  (Big number, how big is the smallest power of two that equals two (too easy, three?) consecutive primes?)

Lagrange's theorem tells us that each positive integer can be written as a sum of four squares (perhaps including zero), but many can be written as the sum of only one or two non-zero squares. 335 is one of the numbers that can not be written with less than four non-zero squares. The smallest examples are 7, 15, and 23. If you take any number in this sequence, and raise it to an odd positive power, you get another number in the sequence, so now you know that 73 = 343 is also not expressible as the sum of less than four non-zero squares.
*Prime Curios

There are 67 primes smaller than 335, and so 335 is divisible by the number of primes less than itself..  How common is that for integers.

There are 335 primes of the form n^8+1 below 10^4.  *Derek Orr

335 = 168^2 - 167^2 = 36^2 - 31^2.

335 = 7^3 - 2^3

The 336th Day of the Year
336 = 24 × 3 × 7,

336 is a Joy-Giver number, divisible by the sum of its digits

336 is the typical number of dimples on an American golf ball.

336 is an untouchable number, it can not be expressed as the sum of all the proper divisors of any other number.

336= 85^2 - 83^2 = 44^2 - 40^2 = 31^2 - 25^2 = 25^2 - 17^2 = 20^2 - 8^2

336 is the smallest number which is the sum in two ways of two primes with indices that are primes, 59 + 277 = 5 + 331 = 336. *Prime Curios

678901234567890123456.... with 336 digits is prime (Largest known number with this property)  *Derek Orr

336^2 + 337^2 + 338^2 is prime.  *Derek Orr

336 = 4^2 + 8^2 + 16^2 = 2^8 + 2^ 6 + 2^4

There are 336 unique ways to partition 41 into prime parts.  (2 + 2 + 37 would be one such, 23 + 13 + 5 would be another.)

336 is the product of three consecutive integers, 6*7*8  = 336

Yesterday I mentioned LaGrange's theorem that every number can be written as the sum of four integral squares. Some can be written as the sum of four squares in many different ways which include the use of 02. The number 26 can be partitioned as the sum of four squares in 336 different ways. If that sounds too trivial, tell folks 336 is a Lipschitz integer quaternion.

The 337th Day of the Year
337 is a permutable prime, every permutation of its digits is prime, 373 and 733 are also prime,  It is the 68th prime number and the last year day which is a permutable prime. There are no three digit permutable primes with all distinct digits.

337 is the last year day which has an inverse that is a maximal digit prime ( 1/n has n-1 digits).  for 1/337 has 336 digits.

337 = 2^8 + 9^2.

337 is a star number like a Chinese checker board , (which has 121 holes) but larger (how big would the home triangles be on such a board.

337 is a Pythagorean prime number.(A Pythagorean prime is a prime number of the form 4n + 1. Pythagorean primes are exactly the primes that are the sum of two squares (and from this derives the name in reference to the famous Pythagorean theorem.)

The mean of the first 337 square numbers is itself a square. This is the smallest number for which this is true.

The famous Fibonacci area paradox shows a 13x13 square converted to an 8x21 rectangle. The areas of the two figures, 13x13 + 8x21 = 337 (this illusion works with any Fibonacci number F(n) squared and a rectangle that is F(n-1) by F(n+1)  )  Students must be aware that 13 x 13 = 169 is NOT equal to 8 x 21 = 168, so where is the flaw.  Here is a post for a little history of these geometric vanishes.

337 = 9^2 + 16^2, and 337^2 = 175^2 + 288^2
337 = 4^4 + 3^4

Jim Wiler at Wilderlab pointed out that 337 = 3 x 3^3 + 4 x 4^3, the sum of  seven third powers.

337 and 373 are both prime, and both concatenations of the factors of 101, 3 x 37 and 37 x 3  *Prime Cuios

The 338th Day of the Year
338 = 2 x 13^2, 338 is the last year day that is twice a square.

338 is the least number, and the only year date, for which the sum of its prime factors, 28, and its number of divisors, 6, are both perfect numbers.  *Prime Curios

338 has a remainder of two when divided by 3, 4, 6, 7, and 8.  (Students might try to create numbers with a remainder of n when divided by three consecutive numbers.)

338 = 4^4 + 3^4 + 1^4
338 is the sum of two squares in two different ways, 13^2 + 13^2 = 7^2 + 17^2.

338 is a happy number, the iteration of sums of squares maps to one.

338 is the arithmetic mean of two triangular numbers.

The 339th Day of the Year
339 = 3*113,  almost a palindrome prime factorization

339 has a remainder of three when divided by 4, 6, 7. and 8.  (compare to 338)

339 is the sum of four 5th powers.  3^5 + 2^5  + 2^5 + 2^5.

It is also the sum of four 4th powers,  4^4 + 3^4 + 1^4 + 1^4
And it is the sum of nine positive 4th powers if you want to seek them.

339 is the fourth, and last, day of the year which can be expressed as the sum of the squares of three consecutive primes. 7^2 + 11^22 + 13^2

The 339th day of the year; the plane can be divided into 339 regions with 13 hyperbolae.

Just discovered the term emirpimes (semiprime reversed) for numbers like 339 and 933 which are semi-primes that are reversals of each other.  933 = 3 x 311, the prime factors are even reversals of each other.

339 = 170^2 - 169^2 = 58^2 - 55^2,

The 340th Day of the Year
340 = 22 × 5 × 17,  Divisible by the number of primes beneath it (337 is the 68th prime)

sum of eight consecutive primes (29 + 31 + 37 + 41 + 43 + 47 + 53 + 59), sum of ten consecutive primes (17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53), *Wikipedia

340 can also be written as the sum of consecutive primes in third way, 167 + 173 *Prime Curios.  Only two more year days can be written as the sum of two consecutive primes.

340! +1 is prime. There are only thirteen day numbers of the year for which n! +1 is prime, and 340 is the last of these.

Jim Wilder@wilderlab pointed out that 340 = 41 + 42 + 43 + 44. Just think, tomorrow will be even a longer string of consecutive powers of four!  (and 340 is the sum of five positive fifth powers)

340^2 + 1 is prime, and 340^4 + 1 is prime. *Derek Orr There is only one more year day for which n^2 + 1 is prime, but two more for which n^4 + 1 is prime.

340 = 4^2 + 18^2 = 12^2 + 14^2.

And for the Geometry folks, 340 is the last year day that a regular n-gon of that number of sides can be constructed with (unmarked) straightedge and compass.

The 341st Day of the Year
341 = 11 x 31

341 is equal to the sum of the squares of the divisors of 16, 1^2 + 2^2 + 4^2 + 8^2 + 16^2

There is no digit that can be inserted on both sides of the 4 such that 3x4x1 is prime.

341 = (4^5 - 1)/3.  Not just a curiosity. numbers in the sequence a(n) = (4^n-1)/3  will collapse to one following the Collatz conjecture after 2n+1 iterations.  so 341 should end in 11 steps, 341 , 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1 (It seems some count iterations differently, OEIS says ends in 2n steps)

341 is the sum of seven consecutive primes, 37 + 41 + 43 + 47 + 53 + 59 + 61

and 341 is also the smallest number with seven representations as a sum of three positive squares (collect the whole set!)

341 is the smallest of the pseudoprimes base 2,  disproving a Chinese math conjecture from around 500 BC. The conjecture was that p is prime IFF it divides 2<sup>2 - 2.

for younger students that really means if you raise two to the 340th  power, and divide by 341, you get a remainder of one.

341 is a palindrome when written in bases 2 (101010101), 4 (11111), 8(525), 17(131), and 30 (bb). *Derek Orr (Only one more year day will be a binary palindrome)

A pseudoprime n in a base b is a composite number n such that$$b^{n-1} \equiv 1 Mod_n$$ for this example, that means that $$2^{341-1} \equiv 1 Mod_341$$ .

Pseudoprimes are also called Poulet numbers, and Sarrus numbers.  "Sarrus numbers" is after Frédéric Sarrus, who, in 1819, discovered that 341 is a counterexample to the "Chinese hypothesis" mentioned above.
"Poulet numbers" appears in Monografie Matematyczne 42 from 1932, apparently because Poulet in 1928 produced a list of these numbers *OEIS

341 is also the smallest number with seven representations as a sum of three positive squares (collect the whole set!)
341=1²+4²+18² 341=1²+12²+14² 341=2²+9²+16² 341=4²+6²+17² 341=4²+10²+15² 341=6²+7²+16²
$$341 = 8^2 + 9^2 + 14^2$$

341 can be written as the sum of five consecutive powers of 4.  (see 340)

341 is also the sum of two (consecutive) positive  cubes, 5^3 + 6^3 = 341 .  Such numbers are called centered cube numbers.  (A centered cube number is a centered figurate number that counts the number of points in a three-dimensional pattern formed by a point surrounded by concentric cubical layers of points, with i points on the square faces of the ith layer.*Wikipedia)

341 = 171^2 -  170^2 = 21^2 - 10^2

The 342nd Day of the Year
342 = 2 x 3^2 x 19, A Joy-Giver, or Harshad, number, divisible by the sum of its digits.

342 = 18 x 19,   is an Oblong (or promic, pronic, or heteromecic) number, the product of two consecutive integers and thus twice a triangular number. There will be no more of them this year. It is the last such day of the year.   (Promic numbers are related to the infinite nested iteration of roots, as I discovered here.)

342 is also the sum of three positive cubes.

OH MY! 342 in base 10 is = 666 in base 7

342^2 = 116964, the concatenation of four squares, 1, 16, 9, and 64.  and it is the product of 2^2 x 9^2 x 19^2 *Derek Orr

342 is the sum of three squares using 3, 3, and 18, or 2, 7, and 17, or 6, 9, and 15.

342^2 + 343^2 + 344^2 is a prime number *Derek Orr

342 is the sum of three positive cubes, 6^3 + 5^3 + 1^3

342 is the tenth term in the Tribonacci sequence starting with 1, 2, 2, 5, 9

The 343rd Day of the Year
343 is a Friedman number (named after Erich Friedman, as of 2013 an Associate Professor of Mathematics and ex-chairman of the Mathematics and Computer Science Department at Stetson University, located in DeLand, Florida *Wik), since it can be made up of arithmetical operations of its digits, (3+4)3 = 343.  There will be one more Friedman number this year; can you find it?

Lagrange's theorem tells us that each positive integer can be written as a sum of four squares (perhaps including zero), but many can be written as the sum of only one or two non-zero squares. 335 is one of the numbers that can not be written with less than four non-zero squares. The smallest examples are 7, 15, and 23. If you take any number in this sequence, and raise it to an odd positive power, you get another number in the sequence, so now you know that 73 = 343 is also not expressible as the sum of less than four non-zero squares.
*Prime Curios

Interestingly, the speed of sound in dry air at 20 °C (68 °F) is 343 m/s.

343 is the smallest cube ending in 3. It is also the last cube of the year. As a perfect cube, it is also a perfect number of the second kind, the product of its aliquot parts is equal to the number itself. In 1879, E. Lionett defined a perfect number of the second kind as a number for which the product of the aliquot parts is equal to the number itself. So 343 is the 7th perfect number of the second kind. The only values that can be perfect numbers of the second kind are values in the form P*Q for primes P, Q, and P3.

Benjamin Vitale@BenVitale noticed that  is a palindrome for n greater than one, you get a palindrome... and if you divide or multiply the result by 7, you get a perfect square. Some simple algebra will allow any HS student to confirm.

343 is the only known example of x2+x+1 = y3, in this case, x=18, y=7.

343 is z3 in a triplet (x,y,z) such that 35 + 102 = 73.  *Wik

343 = 172^2 - 171^2 = 28^2 - 21^2

343 is not prime, in fact, no three digit palindrome with 4 in the middle is prime.  Only one other digit is similarly not found as the middle digit of a three digit prime.

343 is the sum of the first five Odd Fibonacci Primes.  *Prime Curios

Mathematicians of Europe were challenge by Fermat in 1647 to "find a cube, that when increased by the sum of its aliquot parts, is a square."  The cube, was $$7^3 = 343$$ , and the sum of the cube and its aliquot divisors $$7^3 + 1+ 7 + 7^2 = 400 = 20^2$$

The 344th Day of the Year
344 = 2^3  x 43.  That means 344 has 8 divisors, and since it is divisible by eight, it is a refactorable number.  First defined by Curtis Cooper(Prof at U of Central Missouri and co-discoverer of the 43rd and 44th Mersenne Primes) and Robert E. Kennedy (also at U of Central Missouri at the time of discovery), they were later named "refactorable" by Simon Colton (Prof at U of London).

344 is the sum of two positive cubes and of three positive cubes. There will only be one more day for the rest of the year that is the sum of two positive cubes.

The sum of the squares and the sum of the cubes of the prime factors of 344 are both primes, ( and  )  *Prime Curios

344 = 87^2 - 85^2 = 45^2 - 41^2

What does Groundhog Day (Feb 2) have to do with the 344th day of the year?  (Soooo glad you asked!) If you start on New Year's day, and record the Phi function (number of days less than or equal to n and relatively prime to it).  Now on Groundhog day, add them all up.... you get 344. .... Ok, an interesting historical note about what we call the Euler Phi function, Euler used the symbol Pi for it (1784) . Gauss chose the phi symbol(1801), and J J Sylvester gave it the name Totient(1879).

Of the 343 numbers between 344 and 2*344=688, there are 56 primes.  Of the 688 numbers between 344^2 and 345^2  there are 55 primes.  *Derek Orr

The 345th Day of the Year
345 = 3 x 5 x23, a Sphenic or wedge number.

Normally the 345th day happens on Nov 12, or 1112, the fourth "see and say number"  *Derek Orr
1, that's one 1, so 11.  That's two 1's, so 21, and that has one 1 and one 2, so 1112.

345 is the average number of squirts from a cow's udder needed to yield a US gallon of milk. *Archimedes-lab.org (I have not personally verified this, so the proof is left to the reader)

The numbers 345 and 184 form an unusual pair. Their sum is a square, the sum of their squares is a square, and the sum of their cubes is a square.

Jim Wilder@wilderlab pointed out that the digits of 345 show up in two interesting equations, 3^2 + 4^2 = 5^2 and 3^3 + 4^3 + 5^3 = 6^3,

3! + 4! + 5!  +1 = 151, a palindromic prime.  And 345 is the sum of the first six Fibonacci primes, Both from *Prime Curios

345 = 2^8 + 9^2 + 2^3 *PB

345 is the difference of two squares in several ways,  173^2 - 172^2 = 59^2 - 56^2 = 37^2 - 32^2 = 19^2 - 4^2,

345 = 7^3 + 1^3 + 1^3

The Farey sequence using fractions with denominators less that 33 has 345 terms.

The 346th Day of the Year
346 = 2 x 173, There are only two more even semi-primes this year .

346 is a Smith number. The sum of its digits equals the sum of the digits of its prime factors. 346 = 2 x 173 and 3+4+6 = 2+1+7+3. One more such number for a day this year. (Smith numbers were named by Albert Wilansky who noticed the property in the phone number (493-7775) of his brother-in-law Harold Smith.)

346 is also the fourth Franel number, the sum of the cubes of the terms in the nth row of the arithmetic triangle.  The numbers are named for Swiss Mathematician Jérôme Franel (1859–1939).

346 = 11^2 + 15^2

3461, 3463, 3467, and 3469 are all prime.

346 in base nine is a palindrome 424.

The 347th Day of the Year
347 is the 69th prime, the smaller of a pair of twin primes, and a safe prime (also called Sophie Germain primes, meaning that 2 x 347 + 1 is also prime.  There is only one more safe prime this year.  It is also the smallest Friedman prime which is an emirp.

And from Derek at @MathYearRound, "Adding 2 to any digit of 347 keeps it prime (547, 367 and 349 are prime)."

Derek's comment also points out that 347 is the smaller of a pair of twin primes. I just found out that, "(p, p+2) are twin primes if and only if p + 2 can be represented as the sum of two primes. Brun (1919)" (Brun showed that even if there are an infinity of prime pairs, the sum of their reciprocals converges.)

There are 347 even digits before the 347th odd digit of π. (How often is it true that after 2n digits of π there are n even and n odd digits?)

347 is another Friedman number since 347 = 7^3 + 4.  (see 343 for some history notes) It is also the smallest Friedman prime which is an emirp.

347 - (3 + 4 + 7) = 263, a prime, and 347 + (3 x 4 x 7) = 431, another prime.  *Derek Orr

347 preceded by the digits 987654321 is prime.  *Derek Orr

347 - 174^2 - 173^2  = 3^2 + 7^2 + 17^2 = 1^2 + 11^2 + 15^2 = 3^2 + 13^2 + 13^2.

347 is one of the numbers of Euler's incredible sequence of primes of the form n^2 + n + 41, when n = 17.

347 is a left trucatable prime.  Taking off the leftmost digit forms another prime.  347  ---- 47 --- 7.

The 348th Day of the Year
348 is the sum of four consecutive primes. It is the last day of the year that is of such distinction.

348 is the smallest number whose fifth power contains exactly the same digits as another fifth power... find it.

348 =2^2 x 3 x 29 is called a refactorable number because it is divisible by the number of its divisors, (12).

Derek Orr pointed out that 348 + 3 x 4 x 8 and 348 - 3 x 4 x 8 and 348 + 3 + 4 + 8 and 348 - 3 - 4 - 8 are all palindromes.

348 = 88^2 - 86^2 = 32^2 - 26^2

The 349th Day of the Year
349 is a prime, the 70th,  and the sum of three consecutive primes (109 + 113 + 127).

349 is the last day-number of the year that will be a member of a twin prime.

349 is also the largest day-number that is a prime such that p- product of its digits and p+product of its digits are both also prime; for 349, 349 + 3*4*9 = 457 and 349 - 3*4*9 = 241.. and 349, 457 and 241 are all prime. *Ben Vitale

349 was the winning number of the Pepsi Number Fever grand prize draw on May 25, 1993, which had been printed on 800,000 bottles instead of the intended two. The resulting riots and lawsuits became known as the 349 incident.  *Wik

349 is the only prime less than a googol for which 7^p + 6 is a prime (for p a prime).*Prime Curios

349 = 5^2 + 18^2, and 349^2 = 180^2 + 299^2

the sum of 349^n for powers 0 through 6, is prime, *Derek Orr

349 + 3 x 4 x 9 and 349 - 3 x 4 x 9 are both prime

The 350th Day of the Year
350 is S(7,4), a Stirling Number of the second kind.

3502+1 = 122,501 is prime. The last day of the year for which n2 + 1 is prime.

Lucky Sevens, 350 = 73 + 7

Both 350 and 351 are the product of four primes. 350 = 2x5x5x7 and 351 = 3x3x3x13. They are the third, and last pair of consecutive year days that are the product of four primes. (Don't just sit there, find the others!")

350 is a pseudo-perfect number, it is the some of some, but not all, of its proper divisors.  Any multiple of a pseudo-perfect number is also a pseudo-perfect number.  If such a number is not divisible by a smaller pseudo-perfect number, it is called a primitive pseudo-perfect number.   Pseudo-perfect numbers are also called semiperfect.

350 is  also divisible by the number of primes before it, 70.

There are 350 non-square rectangles on a 6x6 grid.  *Derek Orr

The sum of the powers of 350 from zero to six, is prime *Derek Orr

350 is a palindrome in Duodecimal (base 12) 252

350 is expressible as the sum of three squares in three ways; 18^2 + 5^2 + 1^2 = 17^2 + 6^2 + 5^2 = 15^2 +  11^2 + 2^2

The 351st Day of the Year
351 is the 26th  triangular number (27 choose 2), and the sum of five consecutive primes. It is also an element in the Padovan Sequence, an interesting exploration for students. *Wik

351 is the last year date that is a reversible triangular number, since 153 is also triangular.
These two numbers form two of the 11 multi-digit triangular numbers which are divisible by the sum of their digits, 351/9 = 39; 153/9 = 17.

351 can not be written as the sum of three squares.  It is the 85th year day for which that is true, there is only one more this year.  It is also not the sum of two squares.

351 is, however, the difference of two squares in four ways, 351 = 176^2 - 175^2 = 60^2 - 57^2 = 24^2 - 15^2 = 20^2 - 7^2.
and 351 is the sum of two positive cubes, 7^3 + 2^3

351 is the smallest number whose sixth power has six zeros.  (Is there another year day that has this quality?)  *Derek Orr  (he also points out that the first, second, and third powers use no digit greater than five.

351^2 + 2 is prime.
When x=351, x^2 + x +/- 1 form a pair of twin primes.  *Derek Orr

The 352nd Day of the Year
352 is the last day of the year that appears in the Lazy Caterers Sequence, also called the pancake cutting sequence and the Central Polygonal numbers. The numbers describes the maximum number of pieces that a flat disc (or pancake) could be cut with n straight lines. For 26 straight cuts, that number is 352 pieces. The formula is given by P = $$\frac{n^2 + n + 1}{2}$$ .

 *Wik

There are 352 ways to arrange 9 queens on a 9x9 chessboard so that none are attacking another. (Gauss worked on the generalized queens problem; Students might try to find the number for small n x n boards. A general algorithm is not yet known)

352 has all prime digits, and so does the 352nd prime, 2377.

352 = 8^2 + 12^2 + 12^2

352 = 89^2 - 87^2 = 46^2 - 42^2 = 26^2 - 18^2 = 18^2 - 3^2

352 = 173 + 179, the sum of two consecutive primes.

The sum of the digits of 352 divides the product of the digits.  How frequent is this.

The 353rd Day of the Year
353 is the 71st prime number (note that 71 is prime as well) Only one more prime year day this year (or any year).
353 is the last day of the year that is a palindromic prime. It is the first multi-digit palindromic prime with all prime digits.

Also, it is the smallest number whose 4th power is equal to the sum of four other 4th powers, as discovered by R. Norrie in 1911: 3534 = 304 + 1204 + 2724 + 3154. *Wik *R. Norrie, University of St. Andrews 500th Anniversary Memorial Volume, Edinburgh, 1911.

353 is the sum of the first seventeen palindromic numbers, beginning with 0. *Prime Curios

353 is the smallest palindrome that is the sum of eleven consecutive primes, (13+17+19+23+29+31+37+41+43+47+53=353). *Prime Curios

353 = 2^4 + 3^4 + 4^4 *Prime Curios
and similarly, 3^4 + 5^4 + 3^4 = 787, another palindromic prime.  *Prime Curios

353 is the hypotenuse of a Pythagorean Right Triangle.  353^2 = 272^2 + 225^2

353 = 8^2 + 17^2 = 177^2 - 176^2

The 354th Day of the Year
354 = 1^4 + 2^4 + 3^4 + 4^4

354 is  the sum of three distinct primes. (It is also the solution to one version of an unsolved recreational math problem called the Postage Stamp Problem, or sometimes Frobenius problem)

354 = 2 x 3 x 59, a Sphenic number

354 is the smallest number whose sum of the distinct prime factors is a cube, 2+3 + 59 = 64 = 4^3

Of all the Primes less than 10^10, the largest difference between two consecutive primes is 354. *Derek Orr

The 355th Day of the Year
355 is the 12th Tribonacci number, Like Fibonacci but start with 1,1,1 and each new term is the sum of the previous three terms.

355 is almost exactly   No year day is closer to an integer multiple of pi.  For that reason, it offers a really good approximation to Pi, 355/113.  The Chinese often call this ratio Zu Lu after the Chinese mathematician and astrologer, Zu Chongzhi who found it in the 5th Century.

355 is also the last Smith number of the year.  A composite number with the sum of its digits equal to the sum of the digits of it's prime factors 3 + 5 + 5 =  5 + 7 + 1  (355 = 5 x 71)  Numbers where each prime factor is used only once, are called hoax numbers, see 364.

If you write out the binary expression of  355, and examine it as a decimal number, (101100011)  it is prime.  355 is the last day of the year that is such a number.

355 is the last day of the year that is an emirpimeS (semiprime spelled backward) .   Its reversal, 553 is also a semiprime, 7 x 79.

Like 350, 355 is divisible by the number of Primes below it, 71

355 = 178^2 - 177^2 = 38^2 - 33^2,

355 is expressible as the sum of three squares in two distinct ways, 355 = 15^2 + 11^2 + 3^2 = 15^2 + 9^2 + 7^2,

The 356th Day of the Year
When the iterated sum of the squares of digits of a number produce 1, it is called a Happy Number. If any number is Happy, and permutation of it's digits is also Happy, and inserting any number of zeros also will result in a Happy Number (13 for instance is Happy, since 1^2 + 2^2= 10, and 1^2 + 0^2= 1, so 31, 103, 301 and 310 are also Happy Numbers.
356 is happy because it follows the chain 70---49---97---130---10---1
I have proposed the use of the term Principle Happy numbers for those that do not contain a zero, or any reordering of a previous happy number. Here is a list of  the smallest 30 Principle Happy Numbers, and makes searches more direct since no descending sequences of digits can exist. 1, 7, 13, 19, 23, 28,44, 49, 68, 79, 129, 133, 139, 167, 188, 226, 236, 239, 338, 356, 367, 368, 379, 446, 469, 478, 556, 566, 888, 899, Note that the last day year which is a principal happy number is Day 356.  (365 is the last year day that is Happy)

There are 356 ways to partition the number 36 into distinct parts without a unit.

356 is the last day of the year that will be a self-number, (there is no number n such that n+ digit sum of n = 356)

356 = 22 x 89. Numbers that are the product of a prime and the square of a prime are sometimes called Einstein numbers, after E = m c2

356 = 10^2 + 16^2; and 356^2 = 320^2 + 156^2.

356 = 90^2 - 88^2

356 is the sum of three squares in three unique ways.  356 = 18^2 + 4^2 + 4^2 = 16^2 + 8^2 + 6^2 = 14^2 + 12^2 + 4^2

The 356th Prime is 2393.  Note that the concatenation of the two 3562393 is also prime.  Their sum, 356 + 2393 = 2749 is prime as well.  And 3 + 5 + 6 + 2 + 3 + 9 + 3 = 31.... Yep.   *Prime Curios

The 357th Day of the Year
357 is the first three digits of the longest left truncatable prime now known, 357686312646216567629137. If you mark off the leading digits one by one, after each one you still have a prime number, finishing in the prime sequence 9137, 137, 37, 7

There are 357 odd numbers in the first 46 rows of Pascal's Arithmetic triangle. (How many evens?)

357 is made up of three consecutive prime digits, and is the product of three distinct primes, 3 x 7 x 17=357, thus a Sphenic (wedge) number.

There are 21 year dates for which the sum of the divisors is a square number. 357 is the 20th of them. 1+3+7+17+21+51+119+357=576=242

357 is made up of three distinct prime digits, yet it is not a prime number, and none of the five other permutations of the digits is prime. *Prime Curios (Wondering, without checking yet [yeah, I'm lazy] how many three digit numbers with distinct prime digits meet either, or both, of these conditions. Even 253, which has only two permutations that might be prime, has one that is, 523

The 358th Day of the Year
358 is twice a prime, and the sum of six consecutive primes, 47 + 53 + 59 + 61 + 67 + 71

The sum of the first 358 prime numbers is itself a prime number.

and in case you were curious, the 358th digit of pi (after the decimal point) is 3.

358 is smallest number whose first two digits are distinct odd primes, and the third digit is their sum.  *Prime Curios
So I suspect that there is a number with two distinct primes and the third digit is their sum, but one of the primes is two.  Can you confirm, or deny?

35+ 8 + 3 + 58 + 3 + 5 + 8 = 3 x 5 x 8 = 120

Derek Orr pointed out that 358 = 2 x 179, and 2+ 179 = 181 is prime, and 2 + 1 + 7 + 9 = 19 is also prime.

358 = 18^2 + 5^2 + 3^2 = 14^2 + 9^2 + 9^2

The 359th Day of the Year
359 is the 72nd prime of the year, and the last prime year day.

359 is a Sophie Germain prime. If you start with n=89 and iterate 2n+1 you will get a string of primes that includes 359. (How many in all?)

It is also the smallest Sophie Germain prime whose reversal, 953 is also a Sophie Germain prime (so 359 is a  Sophie Germain Emirp, 953 x 2 + 1 gives another Sophie Germain prime, and it is an Emirp as well, unfortunately not with its 2n+1 pair.)
(On most years this year day occurs on Christmas Day, a fitting day for the last prime day of the year .)

e raised to the exponent of (Pi x sqrt(349)) is a 26 digit number that is less than one one-hundredth from being an integer.

The three digit number beginning at digit 359 of Pi, and centered at digit 360, is 360 *Prime Curios.

There are only 88 year days which are not the sum of three non-zero squares.  359 is the last one.
359 is prime, and placing a three between, in front of, or behind, all produce primes.  3359, 3539 and 3593.  *Derek Orr

Like all odd numbers, 359 is the difference of two consecutive squares.  359 = 180^2 - 179^2

359^33 is an 85 digit number whose digit sum is 359.  It is the largest year day I have found for which n^k = digit sum of n.

The 360th Day of the Year
The three digit number centered at digit 360 of Pi, is 360.  However 360 does occur once earlier centered at position 286.]

Bryant Tuckerman found the Mersenne prime M19937 (which has 6000 digits) using an IBM360. *Prime Curios

360 is also the number of degrees in a full circle, and there is a (rather new) word for two angles that sum to 360 degrees.  They are called "explementary" .

360 is a highly composite number, it has 24 divisors, more than any other number of the year, in fact any number that is below twice its size.

It is the smallest number that is divisible by nine of the ten numbers 1-10 (not divisible by 7) What is next, students?

There are 360 possible rook moves on a 6x6 chess board.*Derek Orr

360 = 6^2 + 18^2 , and 360 ^2 = 288^2 + 216^2

360 is divisible by 72, the number of primes below it.

A 360 sided regular polygon is the smallest regular polygon whose angles (in degrees) are prime.  *Prime Curios

360 is also a refactorable or tau number, divisible by the number of its divisors.

The 361st Day of the Year
2361 is an apocalyptic number, it contains 666.
2361=4697085165547666455778961193578674054751365097816639741414581943064418050229216886927397996769537406063869952 That's 109 digits, and as 361 is the last year day that is a perfect square, important to point out for students that all perfect squares are also the sum of consecutive triangular numbers, 361= 171 + 190

There are 361 positions on a Go Board

361 is only the second square year date that requires the sum of  five powers of two to achieve, the first since 121. 361= 2^8 + 2^6 + 2^5 + 2^3 + 2^0

One of Ramanujan's approximations of pi involved 361,  pi is approximately (9 ^2 + 361/22)^ (1/4) = 3.1415926525826461252060371796440223715578779831601261496951353279  *Prime Curios

Derek Orr pointed out that 361 is a square that is the concatenation of two squares, 36 and 1.

361 = 181^2 - 180^2

361 is the sum of three non-zero squares in three ways.  Find them.

361= 19^2  is the largest square year day and is expressible as three consecutive triangular numbers, 361 = 105 + 120 + 136.  (Student note, All squares are the sum of  two consecutive triangular numbers.)

8 x 45 + 1 = 361 = 19^2.  So what?  Well 45 is a triangular number,  and students should be aware that if you multiply any triangular number by 8 and then add one you get a square number.

There are only six year days which are square and not the difference of two primes, 361 is the last one.  All six of the year days with this feature are the square of primes, but for some larger numbers this is not true, both 625 and 1225 are both such squares.

There are 361 terms in the Farey sequence of fractions with denominators less than or equal to 34.

The 362nd Day of the Year
362 and its double and triple all use the same number of digits in Roman numerals.*What's Special About This Number.

3!+6!+2! - 1 =727 and 3!*6!*2! + 1=8641 are both prime *Prime Curios

362 is  the sum of 3 non-zero squares in exactly 4 ways (Collect the whole set)

362 = 1^2 + 19^2 = 17^2 + 8^2 + 3^2 = 16^2 + 9^2 + 5^2 = 15^2 + 11^2 + 4^2

Both 3! + 6! + 2! + 1 and 3! + 6! + 2! - 1 are primes  *Prime Curios

The 363rd Day of the Year
363 is the sum of nine consecutive primes and is also the sum of 5 consecutive powers of three. It is the last palindrome of the year.

363 is the numerator of the sum of the reciprocals of the first seven integers, 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 = 363/140

363 = 182^2 - 181^2 = 62^2 - 59^2 = 22^2 - 11^2

132 plus its reversal, 231, = 363.  This is only the 49th day of the year that is the sum of a integer and its reversal.

363 is only the 45th number that is divisible by the square of its largest prime factor.

The 363rd digit in the decimal expression of Pi is a one.  It is the 37th one in the decimal expression of Pi.

363 = 19^2 + 1^2 + 1^2 =  17^2 +7^2 + 5^2 = 13^2 + 13^2 + 5^2

The 364th Day of the Year
364 = 2^2 x 7 x 13, and is the sum of 12 consecutive primes beginning with 11.

364 = 3^5 + 3^4 + 3^3 + 3^2 + 3^1 + 3^0

364 is a repunit in base 3, 111111, and a repdigit in base 9, 444.

The 364th day of the year; 364 is the total number of gifts in the Twelve Days of Christmas song: 1+(2+1) + (3+2+1) ... which is a series of triangular numbers. The sum of the first n triangular numbers can be expressed as (n+2 Choose 3). The sums of the first n triangular numbers are called Tetrahedral Numbers.  364 is the last year day which is a tetrahedral number.

If you put a standard 8x8 chessboard on each face of a cube, there would be 364*(below) squares. Futility closet included this note on such a cube: "British puzzle expert Henry Dudeney once set himself the task of devising a complete knight’s tour of a cube each of whose sides is a chessboard. He came up with this:

If you cut out the figure, fold it into a cube and fasten it using the tabs provided, you’ll have a map of the knight’s path. It can start anywhere and make its way around the whole cube, visiting each of the 364 squares once and returning to its starting point. (*BTW, I've done the arithmetic on this, and that has to be 384 squares, but I didn't notice the discrepancy at first, so it's still here)

The number of primes less than 364 = 3*6*4 (is this true for any other number?). This product is also the total of the numbers less than 364 which are relatively prime to it.  364 = 2^2 x 3 x 7 and 2 + 3 + 7=13.

364 is the 20th (and last) Hoax number of the year, (the sum of its digits is equal to the sum of the digits of it's distinct prime divisors).  Exactly half those 20 numbers, including this one, have a digit sum of 13.  Numbers that are equal to the sum of the digits of all prime factors with multiplicity (in this number 2 would be added  twice) are called Smith Numbers.  [666 is a Smith number, It's prime factors are 2 x 3 x 3 x 37 and the sum of their digits is 2 + 3 + 3 + (3 + 7) = 18 = 6 + 6 + 6.  Numbers like 22, and 85 are both Smith numbers and hoax numbers. ]

364 is the sum of three squares, 18^2 + 6^2 + 2^2

364 = 92^2 - 90^2 = 20^2 - 6^2

The sum of the  squares of the last three year days in a leap year is a prime, 364^2 + 365^2 + 366^2 is prime.

The sum of the divisors of 364 is a perfect square. It is the 21 st such number of the year.  784 = 28^2.

If you have 11 candles and decide to give them to four children, (that is each child gets 0 - 11 candles and total of all four is 11), there will be 364 possible ways to do it.  The quick answer is 14 choose 3.

The 365th Day of the Year
The 365th (and usually last) day of the year; 365 is a centered square number, and thus the sum of two consecutive squares (132 + 142 ) and also one more than four times a triangular number.

365 is the sum of two squares in two ways, 132 + 142 and  192 + 22 *Lord Karl Voldevive (and of course, it is the largest year day that is the sum of two squares.)

365 like all odd numbers, is the odd leg of a Pythagorean Triangle with a difference of one between the even leg and hypotenuse, 365^2 + 66612^2 = 66613^2

There are 10 days during the year that are the sum of three consecutive squares. This is the last one (proof left to the reader ;-} .
365 = 10²+11²+12²; *jim wilder@wilderlab

365 is the smallest number that can be written as a sum of consecutive squares in more than one way (and all the numbers squared are consecutive.): 365 =102 + 112 + 122 =132 + 142 .

365 is a palindrome in base 2; 101101101
and base 8; 555

365 = 183^2 - 182^2 = 39^2 - 34^2

The 366th Day of the Year
366  is the sum of four consecutive squares, 366 = 8 2,+ 92, + 102, + 112.

366 however is not the sum of two squares.
To check if an arbitrary number is the sum of two squares, factor it.  If any factor p^a + 1 is divisible by four, then it is not a sum of two positive squares.  For 366, the factor of three is the killer.

366 = 2 x 3 x 61 a Sphenic, or wedge number.

366 = 13^3 - 1, which means that in base thirteen it is a repdigit, 222.

366 is the sum of thee squares in three different ways, 19^2 + 2^2 + 1^2 = 14^2 + 13^2 + 1^2 = 14^2 + 11^2 + 7^2

366 is the largest year day for which there is a perfect number with that many digits.  $$2^{606} (2^{607} - 1 )$$