## Monday, 18 March 2013

### MONDAY, 18 MARCH 2013

Today is the $77^{th}$ day of the year.

$77 = 7 \times 11$ which, since $7$ and $11$ are both prime makes $77$ a semi-prime.

$77$ occurs in two primitive Pythagorean triples.
$(77, 36, 85)$
$(77, 2964, 2965)$

$77 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19$ which means that $77$ is the sum of the first $8$ prime numbers, see A007504.

$77! + 1 = 145,183,092,028,285,869,634,070,784,086,308,284,983,740,379,224,208,358,846,781,574,688,061,991,349,156,420,080,065,207,861,248,000,000,000,000,000,001$, which, apparently, is prime. This number consists of $114$ digits. I can understand someone discovering primes. I can understand somebody calculating $77!$. I find it amazing that somebody having calculated of those facts then checked to see if the other one was true. Amazed or not, somebody has checked more than just $77$, see A002981.

Binary Partitions were mentioned on Friday, 15 March 2013 following a short discourse on the more general topic of partitions. To repeat what is stated there:
A partition of a number is a way of writing a number as the sum of positive integers. If two sums contain the same digits and differ only in their order then they are considered the same partition. The number of partitions for a given number $n$ is what we will consider.
As an example consider the number $4$, this can be partitioned in $5$ different ways:
$1) 4$
$2) 3+1$
$3) 2+2$
$4) 2+1+1$
$5) 1+1+1+1$

A little inspection shows that, for the first thirteen numbers, the number of partitions are (see A000041)
$0: 1$
$1: 1$
$2: 2$
$3: 3$
$4: 5$
$5: 7$
$6: 11$
$7: 15$
$8: 22$
$9: 30$
$10: 42$
$11: 56$
$12: 77$

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