Saturday, October 10, 2020

Number Facts for every Year Date, 301-330

 The 301st Day of the year

301 is the sum of three consecutive primes starting at 97

301 = 7 × 43. *Derek Orr points out that 7 = 4+3. Can you find other semi-primes where the sum of the digits of the factors are eqaul? Can you find a product of three primes all with the samed digit sum (sounds like a "work backward" method would be best for both of those.) 

301≡1Modb for every base,b, from 2 through 6 (Sixth grade version, if you divide 301 by any number 2 through 6, you get a remainder of 1)

Any number (such as 301) with prime factors of the form x*(x+36), where x is a prime ending in 7, will end in ...01. E.g., 301 = (7*43), 901 = (17*53), 2701 = (37*73), 3901 = (47*83), etc. *Prime Curios 

301, 302, and 303 are all semi-primes

301, like every odd number is the difference of two squares, 151^2 - 150^2 .  It is also 25^2 - 18^2  (students should expand (x+7)^2 - x^2 to see why, and when this type of relation will next be useful.  

The 302nd day of the year
301, 302, and 303 are all semi-primes

302 = 2 x 151 

302 is a Happy number, it only takes three iterations of the sum of the squares of the digits to get to 1

302^2 = 91204, five distinct digits, and 302^3 = 27543608, with eight distinct digits *Derek Orr, and *PB

The sum of divisors of 302 is 456. 302 written in base 8 is 456 *Derek Orr. Derek also points out that 302 in base nine, is 365. The number of days in a normal year (and 2020 has definitly not been normal.)
There are 302 ways to play the first three moves in checkers.302 is the sum of three consecutive squares 9^2 + 10^2 + 11^2


The 303rd Day of the Year,

301, 302, and 303 are all semi-primes

303 = 3 x 101, 301, 302, and 303 are all semi-primes, and all of them have two factors for which the sum of the digits is a prime, 7 and 7 for 301, 2 and 7 for 302, and 3 and 2 for 303. 

The number of primes less than 2001 is 303. Not impressed, write it the way they did in Prime Curios and you get Pi(10^3 + 10^0 + 10^3) = 303 and it looks much more impressive... "Sell the sizzle, not the steak."

there are 303 different bipartite graphs with 8 vertices. *What's Special About This Number

In the Gregorian calendar, 303 is the number of years that are not leap years in a period of 400 years.
The 304th Day of the Year

304 = 2^4 x 19. Because it has so many factors of two, it is expressible as the difference of two squares in several ways. 
304 /4 = 76 so 77^2 - 75^2 = 304 
because 304/8 = 38, 40^2 - 36^2 = 304 
and because 304 / 16 = 19, 23^2 - 15^2 . 

There are 304 semi-primes less than 2^10, but 304 is NOT one of them. *Derek Orr 

304 is the sum of six consecutive primes (41 + 43 + 47 + 53 + 59 + 61), sum of eight consecutive primes (23 + 29 + 31 + 37 + 41 + 43 + 47 + 53), *Wik 

304 is the record number of wickets taken in English cricket season by Tich Freeman in 1928, (and I do hope they got them all back!)

Math Joke for Halloween: Why do mathematicians confuse Halloween and Christmas? Because Oct 31 = Dec 25 (31 in base 8 (Octal) is the same quantity as 25 in Decimal)

  
304 (3! * 0!   * 4!) + 1 = 43777, a prime with a prime length.  *Prime Curios

The 305th Day of the Year 
305 = 5 x 61, another semi-prime, along with 301, 302, 303, and now 305.  Like the others, the two factors have a sum of digits that are prime, 5 and 7.  

305 is the smallest odd composite which is the average of two consecutive Fiboancci numbers *Number Gossip

305 has two representations as a sum of two squares, 305 = 4^2 + 17^2 =  7^2 + 16^2

and like most numbers that end in 5, 305 is expressible as the difference of two squares.  305 = 33^2 - 28^2.  Algebra students should check which numbers ending in five are not the difference of squares, and how to find out the two numbers without guess and test. 


The 306th Day of the Year 
306 = 2 × 32 × 17. 

306 = 9^2 + 15^2
306 is the sum of four consecutive primes (71 + 73 + 79 + 83)

306 is a pronic number, the product of two consecutive natural numbes, (16 x 17 ) which is twice the 16th triangular number.

The 306th day of the year; 306 is the sum of four consecutive primes starting with 71.

There are 306 triangular numbers with five digits. (students, how many triangular numbers have 3 digits... Can you calculate without listing them all?)

306 = 92 + 92 + 122; (not impressed?) you can write the same numerals with exponents and 306 =92 + 92 + 122 Really? Still not impressed, how about 306 = 82 + 112 + 112= 82 + 112 + 112.

3*306 + 0*306 + 6*306 - 1 is prime *Prime Curios

There are 306 primes less than 45^2.

306 + 1 is prime
306^2 + 1 is prime.
306^8 + 1 is prime.
306^16 + 1 is prime.
The 307th Day of the Year 
The 307th day of the year; 307 is the last day of the year whose square is a palindrome, 3072=94249. The next number whose square Is a palindrome is 836. 

306 is the only number less than 100,000 with a palindromic square with an even number of digits; 8362=698896 Another Ambigram Palindrome, 180 degree rotation becomes 968869

The smallest number that is the sum of any set of primes containing all digits 0-9 : 2 + 5 + 41 + 67 + 83 + 109 = 307.

The largest number that you can type in Excel is 9.999 * 10307. If you type in a larger number, Excel will treat it as a character string. *Prime Curios (Is this still true? Afraid I don't use Excel much anymore.)

307 is the smallest number that is the sum of any set of primes containing all ten digits: 2 + 5 + 41 + 67 + 83 + 109 = 307. 
,br> Smallest of the first case of seven consecutive primes that remain prime if you eliminate their first digit. 
What??? Ok, so 307, 311, 313, 317, 331, 337, 347, are all still primes if you drop the three in front. 

307 is prime, it and its prime index (63) are both reverse triangular numbers. Are there any more examples of this property? *Prime Curios *Prime Curios (computer search anyone?)
307 = 2^4 x 3^3 - 5^3 .  Is it possible to write every prime as a in the form p^a x q^b +/- r^c where p, q, and r are one each (in any order) of 2, 3, and 5?  Gary Croft demonstrated all the ones under 100, 

The 308th Day of the Year 
308 is divisible by four, so 78^2-76^2 = 308 
  
308 is the sum of two consecutive primes.

If 18 circles are drawn in the plane, they can separate the plane into 308 regions. Student's might try to find the maximum number of regions for smaller numbers of circles, find a pattern, write f(n).

3083 + 3080 + 3088 is prime *Prime Curios

Derek Orr noticed that 308 times the sum of its factors, 308 (2+2+7+11) = 6776, a palindrome; and if you multiply by the sum of the digits of its factors, 308(2 + 2 + 7 + 1 + 1) =4004, you get another palindrome. 

308 = 78^2  - 76^2

Derek also pointed out that there are 308 primes of the form x^4 + 1 that are smaller than 10^14,  (that's the answer, but who would think to ask the question?)

The 309th Day of the Year
3095= 2,817,036,000,549. It is the smallest number whose fifth power contains all the digits 0 to 9. (Students, is there a smaller number that contains all the digits for some other power?)

309 is the fifth semi-prime between 300 and 310.  It is the only one for which the sum of the digits of its prime factors are not both primes.  

if x=309, then x^2 + x + 1 is prime.   *Derek Orr; how many year days make this quadratic a prime. 1, works, 2 works... now you works....(tee hee)

The 310th Day of the Year

310 = 1234 in base six. In base 2 it repeats one period, 100,110,110

(1!)²+(2!)²+...+(310!)² is prime. *Math Year-Round ‏@MathYearRound

The four possible 3 digit permutations of 310 all use the same digits in their squares as well, 1302 = 16900, 
3102 = 96100, 1032 = 10609,and 3012 = 90601.

310 in base 6 is 1234.  *Derek Orr


The 311 th Day of the Year 
311 is the only year day, and thus the smallest prime number which is the sum of three, five, and seven consecutive primes. (of course, go find them!)

311 is the eleventh three-digit prime for which the sum of the squares of its digits is also a prime. Note that the sum here is eleven as well.  *Prime Curios

 311 is the smallest number expressible as the sum of consecutive primes in four ways. It can be expressed as a sum of consecutive primes in four different ways: as a sum of three consecutive primes (101 + 103 + 107), as a sum of five consecutive primes (53 + 59 + 61 + 67 + 71), as a sum of seven consecutive primes (31 + 37 + 41 + 43 + 47 + 53 + 59), and as a sum of eleven consecutive primes (11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47).

The smallest three-digit prime that is the sum of three different three-digit primes. (101 + 103 +  107 = 311).  *Prime Curios

311 is the 64th prime; a twin prime with 313; *Wik

311 is a strictly non-palindromic number, as it is not palindromic in any base between base 2 and base 309. *Wik

Every permutation of 311 is prime, 31, 13, 11, 113, 131, 311,  *Derek Orr

The digit sum of 311 is 5, the digit sum of 311^2 is 25  *Derek Orr

311 is also smallest prime after 2, in a set of five consecutive primes each whose sum of digits is prime.  (311, 313, 317, 331, 337)

The 312th Day of the Year
the number is expressed 2222 in base five. 

[It is also a Zuckerman number (a number that is divisible by the product of its digits  (which is the same as the sum of its digits making it also a Harshad (joy-giver) number *PB)) Thanks to David Brooks].  312/6 = 52

312 = 2^3 x 3 x 13.  Because it is divisible by a third power of two, it is expressible as the difference of two squares in two ways, 79^2 - 77^2 = 312, and 41^2 - 37^2 = 312

Derek Orr points out that the prime factors of 312, and 312 itself use only the digits 1, 2, and 3.

312 = 0!*5! + 1!*4! + 2!*3! + 3!*2! + 4!*1! + 5!*0! *Derek Orr

and 3122 is the sum of the cubes of the integers from 14 to 25. 3122=n=1425n3 *archimedes-lab.org

312 is between two twin primes 311 and 313.  Prime Curios points out that the "see and say" number of 312 shares this quality, that is 131112 (one three, one one, and one two) is between a pair of twin primes as well .
Patrick Honaker says this is the smallest such example.  

312 can be written as the sum of three squares in only one way, 4^2 + 10^2 + 14^2

Derek Orr also says that there are 312 ways to place 7 non-attacking queens on an 7x8 board.  (Derek, nobody plays chess on a 7x8 board.)

Derek also adds that in base 7 and base 11, 312 uses the same digits, 624 and 264. 

And in a motion of cryptic chicanery, Derek says that for bases 18 through 36, 312 uses a letter in all but one of those bases, and challenges the reader to find the unusual which is all numerical digits.

The 313th Day of the Year
313 is prime, and 3313 is prime, 33313(7*4759) is not, but 33311 is. Going the other way, 3133 is not, but 31333 is prime, and 313333 is prime, 5 threes on the end is not, six threes on the end is. Pursue on your own, like (see day 331 for a long sting of such primes)

 a twin prime with 311. If you draw all the diagonals of a regular dodecagon, it has 313 intersections.(I believe this is counting the 12 vertices of the dodecagon as well as 301 interior intersections.)

bonus fact: 313 is the only 3-digit palindromic prime that is also palindromic in base 2: 100111001 * Mario Livio

And thanks to the folks at The Zoo of Numbers at Archimedes Lab, I now know that 313 is Donald Duck's License plate number. (But I have also seen Disney authorized materials with license DD-13)

313 is the sixth multidigit palindromic prime, and the last which is a year day.  (11, 101, 131, 151, 181, 313,)

313 is the 65th prime number, and the larger of a prime pair with 311. This is the seventh prime to have a sum of digits of seven . *Prime Curios

313 is the sum of two squares, 12^2 + 13^2  , and is the hypotenuse of a Pythagorean triangle, 25^2 + 312^2 = 313^2.  

313 = 2^5 x 3^2 + 5^2 .  Is it possible to write every prime as a in the form p^a x q^b +/- r^c where p, q, and r are one each (in any order) of 2, 3, and 5?  Gary Croft demonstrated all the ones under 100,  Also see 307 on this page.

Frenicle challenged Wallis to solve 113 x^2 + 1 = y^2.   

313 is the smallest three-digit prime that is not the sum of consecutive composite numbers. *Prime Curios

313 is the sum of the first 63 digits of pi.  The sum of the first n digits is prime for only 11 year days.  The largest or eleventh, is the sum of the first 63 digits of pi, 3 + 1 + 4 + 1 .....  


313 is the sum of 144, the largest Fibonacci square, and 169, the largest Pell square

The 314th Day of the Year
314 is the first three digits of Pi

The 314th day of the year; 314 is the smallest number that can be written as the sum of of 3 positive distinct squares in 6 ways. *What's Special about this number
(Students, can you find the smallest number that can be written as the sum of two distinct squares in at least two ways?)

314 is a semi-prime (2 x 157), and 3142 +1 is prime
Another semi-prime whose factors both have a sum of digits that are both prime. 

314 = 5^2 + 17^2

314 is 222 in duodecimal 

x^2 + x + 1 is prime if x = 314 (and some others)  


The 315th Day of the Year
315 = 3^2 x 5 x 7  (or 5 x 7 x 9)  

Since 315 is divisible by the sum of its digits, it is a Harshad, or Joy-giver number.  Its also divisible by the product of its digits, or a Zuckerman number.   There are only 13 three digit numbers that have both these qualities combined, and 9 of them are year days.  315 is the largest year day with this property.

3152 can be written as the sum of the cubes of five consecutive integers. Find them. (Students may also wish to find the smallest square that can be expressed as the sum of the cubes of two or more consecutive integers.)

Remember those pictures you see where they ask you how many triangles in a picture like this one"
Well the sequence of numbers depends on how many triangles on a side.  Oeis gives the sequence as 
1, 5, 13, 27, 48, 78, 118, 170, 235, 315 and thus with ten toothpicks on a side of the largest triangle, 315 is the largest year day which answers the problem of "How Many Triangles?" In the one shown above, search for the 16 triangles with one unit sides (10 pointed up, six down), then 7 with two unit sides (6 up one down), then 3 with three unit sides and one with four unit sides... for 27).

315 is a (barely)deficient number, the sum of it's proper divisors is only 309 ...309/315 is about .9809 *Derek Orr @Derektionary  pointed out to me that 256 is the closet to one of any (non-perfect) year day, (255/256 = .996), and for non-powers of two, 136 with a ratio of 134/136 = .985 is the best.

Derek Orr pointed out that 315 times its reversal, 513 forms a six digit number that is the concatenation of two palindromes, 161595

On a non leap year, the 315th day is 11/11 

315 is the difference of two squares in more than one way.  158^2 - 157^2 and 26^2 - 19^2 and 54^2 - 51^2,   The first is a property of all odd numbers, For a clue to the second, see Day 301 , For the third, think of how it is similar to the second.  

 There are only 13 three digit numbers that have this property, and 9 of them are year days.  315 is the largest year day with this property.

The 316th Day of the Year
316 = 2^2 x 79

316 = 7^3 - 3^3 = 80^2 - 78^2 

316 can be written as the sum of 3 consecutive triangular numbers.

316 is also a centered triangular number (these two definitions turn out to be identical, all sums of 3 consecutive triangular numbers are centered triangular numbers, and vice-versa) The nth Centered triangle number has a single point at the center (the first centered triangular number) , and triangles surrounding it with 2,3,..., n dots on a side. The fourth centered triangular, 19 =3+6+10, number is shown



Students often enjoy finding cycles in the "sort then add" sequence. 316 is the only year date that does not reach a sorted number to terminate(at least we haven't found it yet) . [If a number is not sorted, then add n to sorted n, for example 21 is not sorted, so f(21)= 21+12=33  (a sorted number) 65 is not sorted, so f(65)= 65+56 = 121, still not sorted so do 121+112= 233, a sorted number so f(65) terminates in 233]

Derek Orr pointed out that there are 316 primes of the form x^2 -1 below 10,000,000

Derek also pointed out that 316 in base 7 uses the same digits, 631.

The 317th Day of the Year 
317 is the 66th prime number. The number made up of 317 consecutive ones digits is also prime. It is the fourth prime repunit. The smallest is 11. Find the other two. *Wik  John Carlos Baez points out that, "That's not a coincidence. A number whose digits are all 1 can only be prime if the number of digits is prime! This works in any base, not just base ten. Can you see the quick proof?"

317 is the only prime year date, p, such that 2^p + p is prime.  The prime is 96 digits long.
 
Derek Orr pointed out that that number with 317 ones, is (10^317 -1)/9 

Not only is 317 the sum of two squares, (11^2 + 14^2)  but 3172 is also the sum of two squares (75^2 + 308^2,  making 317, 308, 75 a Pythagorean Right triangle. Can you find other primes for which both these conditions are true?

317 is the largest year day which is a Russian Doll Prime (Right Truncatable Prime). A number which remains a prime as each rightmost digit is stripped away, 317, 31, 3. There are only a total of 83 such numbers, the largest of which is the eight digit 73,939,133.

If you square the digits of 317, you get 59, together you have all the odd digits.  *Prime Curios

\(317 = (-3)^3 +1^3 + 7^3 \) *Prime Curios

Derek Orr recognized that 317 is the concatenation of two primes in two different ways, 31, 7 and 3,17.  

317 has a remainder of two when divided by 3, 5, 7 or 9

The 318th Day of the Year

According to Police chief Wiggum on The Simpsons; 318 is the Police code for waking a police officer in episode 5F06 Reality Bites.

If 22 is partitioned into distinct integer parts in all possible ways, there will be 318 total parts.

318 is a palindrome in base 9, 3839

318 = 2 x 3 x 53, a sphenic number (from the Greek for "wedge")

318 in base three is a double digit palindrome, 10 22 10

318 has a remainder of three when divided by five, seven, or nine

The 319th Day of the Year
319 = 11 x 29, another semi-prime with the sum of the digits of its factors are both prime, 2 and 11.  
It is the seventh semi-prime since 300, and six of them have the sum of the digits of their factors all prime.  

319 is the sum of three consecutive primes (find them). 

319 also is the largest number whose cube has all distinct digits 3193 =32461759. What is the largest square with all distinct digits?

319 cannot be represented as the sum of fewer than 19 fourth powers: 319 = 3 x 34 + 4 x 24 + 12 x 14

There are six ways to concatenate the number 319 and its factors, 11 and 29.  Each of them is a semi-prime as well.  *Prime Curios

The sum of the digits of 319 is 13, the sum of the digits of its prime factors, 11 and 29, is also 13.  A Smith Number.

And 319 is a Happy number.  The iteration of the sum of the squares of its digits leads to one,  \(3^2 + 1^2 + 9^2 = 91\); \( 9^2 + 1^2= 82\); \(8^2 + 2^2 = 68\);  \(6^2 + 8^2 = 100\), and \(1^2 + 0^2 + 0^2 = 1\).

 319 is another that has the same remainder when divided by 5, 7 or 9, each having a remainder of 4.

Derek Orr added that the first prime after 10^29 is 10^29 + 319 


The 320th Day of the Year
320 is the ninth Leyland Number.  A number of the form x^y + y^x where both x and y are greater than one. \(8^2 + 2^8 = 320\).  Named for Paul Leyland, a British number theorist.

 320 is the maximum value of the determinant of a 10x10 binary matrix (all entries are either one or zero). (Students might explore all possible determinants of smaller matrices looking for a pattern)

320!+1 is prime.

4^2 + 8^2 = 80, so 8^2 + 16^2 = 320.  

320 = 2^6 x 5.  With so many factors of two, it has to have several expressions as the difference of two squares,  \(81^2 - 79^2 =  42^2 - 38^2 = 24^2 - 16^2 =  18^2 - 2^2  = 320\)

320! + 1 is prime, *Derek Orr

The 321st Day of the Year
321 = 3 * 107

321 is the number of partitions of 13 into at most 4 parts. (7+3+2+1 would be one such)

321 is a Central Delannoy number. The Delannoy numbers are the number of lattice paths from (0, 0) to (b, a) in which only east (1, 0), north (0, 1), and northeast (1, 1) steps are allowed. Central Delannoy numbers are paths to (a,a). [Delannoy numbers are named for Henri Auguste Delannoy (1833–1915) who was a friend and correspondent of Edouard Lucas, editor of Récréations Mathématiques.]

\(e ^ (\Pi * \sqrt{321})\) = 2784914870820244444545897.963......  almost an integer

3-2-1, and the sum of the divisors is 4-3-2.  *Derek Orr

3! -2! +1! and 3! -2! -1! are both prime  

Using the digits of 321 exactly once each, it is possible to express  two different numbers as the sum of primes, 2 + 13 = 15; and 2+31 = 33. (You can, of course, do the same with all permutations of 1, 2, 3. 

567^2  = 321489  using all nine non-zero digits. @fermatslibrary

Derek Orr pointed out that the digit 8 appears 321 times in all the four digit primes (where does he get these things??)

The 322nd Day of the Year
322 = 2 x 7 x 23  another sphenic or "wedge" number  which means that it is the area of  a rectangular box (parallelepiped) with prime lengths for its length, width, and height.  The sphenoid bone in the skull is so named because it is essentially a box like shape, but has two hollow sinus cavities.

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

322 is the 12th Lucas Number. The Lucas Sequence is similar to the Fibonacci sequence with L(1) = 1 and L(2) = 3 and each term is the sum of the two previous terms. L(n) is also the integer nearest to ϕn This is the last day of the year that will be a Lucas Number.

322 is smallest number whose square has 6 diff digits (103684). *Derek Orr

\(3^n + 2^n + 2^n\) is prime for n = 0, 1, 2, 3, 4, 5, and  6.  *Prime Curios (surely you wonder if there is another power greater than six for which this is prime... SURELY!)

321, 322, and 323 form the sides of an almost equilateral triangle.  

322 can be written as the sum of three squares in two different ways.  
\(3^2 + 12^2 + 13^2 = 4^2 + 9^2 + 15^2 = 322\)


The 323rd Day of the Year 
323 is a palindromic semi-prime with all prime digits that is the product of a twin prime pair, 17 x 19 Prime Curios says it is  the smallest such palindrome.  No other such palindrome is known.  

If you drew every possible path from (0,0) to (8,0) that never dropped below the x-axis using only unit vectorial moves with slopes of 1, 0, or -1 there are 323 possible paths. (alternatively this is the number of different ways of drawing non-intersecting chords on a circle with eight points- this is deceptive because it counts each way of drawing a single chord, and drawing no chords at all, students might want to count how many ways this can be done using four chords.) These are called Motzkin numbers, after Theodore Motzkin.
  This is the eighth such number.

323 is the sum of nine consecutive primes 323 = 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 *Derek Orr
It is also the sum of 13 consecutive primes, (5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47) *Wik

323 is a palindrome and also the smallest composite number n that divides the (n+1)st Fibonacci number. *What's Special about this Number

The concatenation of 323 with itself, 323323 = 7 x 11 x 13 x 17 x 19, five consecutive primes. The sum of the squares of those primes is 989, another palindrome. *Prime Curios

The sum of the digits of its prime factors (1 + 7 + 1 + 8) is the same as the product of its digits )3 x 2 x 3.)  *Derek Orr

Derek Orr also pointed out that inserting a 9 between any two digits of 323 forms a prime, 3923 and 3293.  He didn't mention it, but 39293 is also prime. 

The 324th Day of the Year 
324 = 2^2 x 3^4 = 4 x 3^4 = 18^2

324 is the sum of four consecutive primes (73 + 79 + 83 + 89)*Wik

324 is divisible by the sum of its digits, and thus a Harshad or Joy-Giver number.  

An untouchable number is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer, 324 is the smallest untouchable number which is a perfect square.  *Prime Curios

324 is the largest possible product of positive integers with a sum of 16. (Students, Can you find the integers. Try to find the similar maximum product with a sum of 17)).


If you have a square array of 324 dots (that's 18x18) you can carefully paint them each in one of four colors so that no four corners of a rectangle (with sides horizontal and vertical) are the same color. you can also do that for any smaller square, but not for any larger. Here is a 17x17 to ponder

324 times its reversal, 423, is equal to 137052, all the non-composite digits.  *Derek Orr

324 in base 2 combines the first three powers of ten, 101000100.

The 325th Day of the Year
325 = 5^22 x 13, the 25th Triangular number.
(Student Note:  multiply any triangular number by 9 and add 1, you get another triangular number.  9 x 36  + 1 = 325.  )

325 is the smallest number that can be expressed as the sum of two squares in three different ways,  \(325 = 1^2 + 18^2 = 6^2 + 17^2 = 10^2 + 15^2 \)

325 is last year day that is the sum of the first n^2 integers, 

325 is a palindrome in base 2, 101000101; in base 4, 11011; and in base eight, 505. 

On an infinite chessboard, there are 325 different squares that can be reached in 5 knight moves.

325 has a remainder of one when divided by 2, 3, 4, 6, 9 or 12

For the numbers less than 10^6, there are 325 number n for which n, n+6, n+12, and n+18 are all prime.  *Prime Curios

325 is the hypotenuse of two Pythagorean Triangles.  (36, 323, 325 and 204, 253, 325)

325 is the smallest triangular number that is the average of the squares of a pair of twin primes, 17 and 19.  (Bet you are wondering what the next one is... huh, aren't you?..... well???)

3251, 3253, 3257, 3259 are all prime.  *Derek Orr

and Derek also has 325 is a palindrome when written in base 2, 4,  8, 18, and 24

The 326th Day of the Year
326 = 2 x 163, a semiprime.  Of the nine numbers from 321 to 329, six are semi-primes.  

326 x ( 3 x 2 x 6) = 362 a reordering of its digits.  *Derek Orr

An interesting combination of two statements from Derek Orr, students should consider the mutual implications:
A)   Of all the the four digit primes there are 326 fours,
B)  There are 326 primes less than 10,000 with at least one four.  

326 is the maximum number of pieces that may be produced in a pizza with 25 straight cuts. These are sometimes called "lazy caterer numbers" and more generally they are centered polygonal numbers.

326 is also the sum of the first 14 consecutive odd primes: 326 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47. *MAA

326 prefixed or followed by any digit still remains composite.  *Derek's Daily Math  I would think such a number might be called an anti-prime .   Wondering if there are two digit  anti-primes? (seems both 20 and 32 are easy examples) 
How many primes can be formed with n326m (with n and m not necessarily distinct) ? I found 13263 is prime.  How many are there in all.  

326 is a palindrome in base 12, 232.

Unless I missed something, 326 is the smallest number missing in the Prime Curios listing.  

Perhaps this one from Derek Orr  can capture their interest, the 54th Prime is 251, and the 54th composite is 75.   SO WHAT?   251 + 75 = 326.  

The 327th Day of the Year
327 = 3 x 109

327 is the largest number n so that n, 2n, and 3n together contain every digit from 1-9 exactly once. (Students might search for a smaller number with that quantity) *What's Special About This Number

and from Jim Wilder @wilderlab:
For day 327: 327 is a perfect totient number- 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.

The number 327 in base ten is equal to \( 57_{[64]} \) but also \( 75_{[46]} \) 

327 cannot be written as the sum of three squares.   Gauss found that this is only true for numbers of the form 4^k (8n-1), such as 7, 15, 23, .... but also 28, 60, 92, ...  and 112, 240, etc.

327 is not prime, but  one in front, 1327, or a one in the back, 3271 makes it prime, but not both.  

There are 327 primes less than three to the seventh (3 2 7) *Derek Orr

In the Collatz (or 3n+1) sequence, no year day takes so long as 327, which takes 143 steps to reach 1.

The 328th Day of the Year
328 = 2^3 x 41,  

Another number that is three powers of ten in binary, and this one in order, 101001000.  

328 is the sum of the first fifteen primes. It is the last year day that is the sum of the first n squares.

It is also is a tau-number since it is divisible by the number of divisors it has.

\(328 =18^2 + 2^2 \) and
\(328 = 6^2 + 6^2 + 16^2\)

328 reversed is prime, but 823 can't be written as the sum of two squares, or as the sum of three squares.

3280123456789 is prime  *Derek Orr

328 can be written as the difference of two squares as well, and in two different ways.  \(328=83^2 - 81^2 = 43^2 - 39^2\)

The 329th Day of the Year
329 = 7 x 47, the sixth of the last nine numbers that is a semi-prime.

329 is not divisible by 3, 2, or 9, but on the bright side, the sum of all its proper divisors is a palindrome, 55.

329 is the sum of three consecutive primes. 107 + 109 + 113

329 is the number of forests (trees and disconnected trees) possible with ten vertices.

329 is also a happy number, since the iteration of the sums of the squares of its digits includes 1, 

329 is another value of n for which n^2 + n + 1 is prime.  Computer people, how many year days are like 1, 2, 3, and 329 and make that quadratic prime?

Teachers, a great day to remind your students why a 4n+1 day like 329 is not expressible as the sum of two squares.

329 is expressible as the sum of three squares in three unique ways, \(18^2 + 2^2 + 1^2 =  17^2 + 6^2 + 2^2 = 10^2 + 15^2 + 2^2 =16^2 + 8^2 + 3^2 = 329 \) , and more ...

Like all odd number, 329 is the difference of consecutive squares, \(165^2 - 164^2\), but it is also \(27^2 - 20^2\). 

The 330th Day of the Year
330 = 2 x 3 x 5 x 11,  and the sum of six consecutive primes, 43 + 47 + 53 + 57 + 61 + 67, and of five consecutive squares 

If all the diagonals of an eleven sided regular polygon were drawn, they would have 330 internal intersections.

330 is the last year day which is a pentagonal number. It is the sum of fifteen consecutive integers starting with the integer 15. (All Pentagonal numbers follow a similar pattern) The average of all the pentagonal numbers up to 330 is the 15th triangular number.

A set of 11 points around a circle provide the vertices for 330 quadrilaterals.  Thus 11 choose 4 = 330

330 is divisible by the sum of its digits, making it a Harshad, or Joy-Giver number.

The sum o 330 raised to each of its digits in turn, is a prime, \(330^3 + 330^3 + 330^0\) is prime.  *Prime Curios

330 = 303 + 3^3  *Derek Orr

And who would have thought, there are 330 dimples on a British Golf Ball, *Derek Orr, 
It seems most US golf balls have 336, but some manufacturers have as many as 360, and one has as many as 1120.  the 2017/18 model of the popular Titleist Pro V1 has 352 dimples on it, (Think they will send me a case?)

330 is the sum of three squares in multiple ways, \(4^2 + 5^2 + 17^2 = 5^2 = 7^2 + 16^2 = 330\)


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