Tuesday, 26 MARCH 2013

Today is the $85^{th}$ day of the year.

$85 = 5 \times 17$

$85 = 2^2 + 9^2 = 6^2 + 7^2$, so it is the sum of two squares in two different ways.

$85$ is the member of the following four primitive Pythagorean triples:

$(13, 84, 85)$

$(77, 36, 85)$

$(85, 132, 157)$

$(85, 3612, 3613)$

$85$ is a Joke Number or a Smith Number. A Joke or Smith Number is one where the sum of the digits of the number, $8 + 5 = 13$, is equal to the sum of the digits of the factors of that number $5 + 1 + 7 = 13$, see A006753.

$\frac {4^4 - 1} {3} = \frac {256 - 1} {3} = \frac {255} {3} = 85$ so $85$ is a member of the sequence $\frac {4^n -1} {3}$, see A002450. What I find surprising is that this sequence implies that all powers of 4 less one are a multiple of three.

It transpires that there are $85$ different ways partitions of $29$ into at most three parts, see A001399.
The $85$ partitions are:

$1: [0, 0, 29]$
$2: [0, 1, 28]$
$3: [0, 2, 27]$
$4: [0, 3, 26]$
$5: [0, 4, 25]$
$6: [0, 5, 24]$
$7: [0, 6, 23]$
$8: [0, 7, 22]$
$9: [0, 8, 21]$
$10: [0, 9, 20]$
$11: [0, 10, 19]$
$12: [0, 11, 18]$
$13: [0, 12, 17]$
$14: [0, 13, 16]$
$15: [0, 14, 15]$
$16: [1, 1, 27]$
$17: [1, 2, 26]$
$18: [1, 3, 25]$
$19: [1, 4, 24]$
$20: [1, 5, 23]$
$21: [1, 6, 22]$
$22: [1, 7, 21]$
$23: [1, 8, 20]$
$24: [1, 9, 19]$
$25: [1, 10, 18]$
$26: [1, 11, 17]$
$27: [1, 12, 16]$
$28: [1, 13, 15]$
$29: [1, 14, 14]$
$30: [2, 2, 25]$
$31: [2, 3, 24]$
$32: [2, 4, 23]$
$33: [2, 5, 22]$
$34: [2, 6, 21]$
$35: [2, 7, 20]$
$36: [2, 8, 19]$
$37: [2, 9, 18]$
$38: [2, 10, 17]$
$39: [2, 11, 16]$
$40: [2, 12, 15]$
$41: [2, 13, 14]$
$42: [3, 3, 23]$
$43: [3, 4, 22]$
$44: [3, 5, 21]$
$45: [3, 6, 20]$
$46: [3, 7, 19]$
$47: [3, 8, 18]$
$48: [3, 9, 17]$
$49: [3, 10, 16]$
$50: [3, 11, 15]$
$51: [3, 12, 14]$
$52: [3, 13, 13]$
$53: [4, 4, 21]$
$54: [4, 5, 20]$
$55: [4, 6, 19]$
$56: [4, 7, 18]$
$57: [4, 8, 17]$
$58: [4, 9, 16]$
$59: [4, 10, 15]$
$60: [4, 11, 14]$
$61: [4, 12, 13]$
$62: [5, 5, 19]$
$63: [5, 6, 18]$
$64: [5, 7, 17]$
$65: [5, 8, 16]$
$66: [5, 9, 15]$
$67: [5, 10, 14]$
$68: [5, 11, 13]$
$69: [5, 12, 12]$
$70: [6, 6, 17]$
$71: [6, 7, 16]$
$72: [6, 8, 15]$
$73: [6, 9, 14]$
$74: [6, 10, 13]$
$75: [6, 11, 12]$
$76: [7, 7, 15]$
$77: [7, 8, 14]$
$78: [7, 9, 13]$
$79: [7, 10, 12]$
$80: [7, 11, 11]$
$81: [8, 8, 13]$
$82: [8, 9, 12]$
$83: [8, 10, 11]$
$84: [9, 9, 11]$
$85: [9, 10, 10]$