### Fifteen

31May11

Fifteen is a triangular number, a hexagonal number, a pentatope number and the 4th Bell number. Fifteen is the double factorial of 5. It is a composite number; its proper divisors being 13 and 5. With only two exceptions, all prime quadruplets enclose a multiple of 15, with 15 itself being enclosed by the quadruplet (11, 13, 17, 19). 15 is also the number of supersingular primes.

15 is the 4th discrete semiprime (3.5) and the first member of the (3.q) discrete semiprime family. It is thus the first odd discrete semiprime. The number proceeding 15; 14 is itself a discrete semiprime and this is the first such pair of discrete semiprimes. The next example is the pair commencing 21.

(Yes, yes all of this is stolen from elsewhere – Wikipedia to be specific so it’s probably all erroneous anyway, but there is a point.)

The aliquot sum of 15 is 9, a square prime 15 has an aliquot sequence of 6 members (15,9,4,3,1,0). 15 is the fourth composite number in the 3-aliquot tree. The abundant 12 is also a member of this tree. Fifteen is the aliquot sum of the consecutive 4-power 16, and the discrete semiprime 33.

15 and 16 form a Ruth-Aaron pair under the second definition in which repeated prime factors are counted as often as they occur.

Fifteen is the magic constant of the unique order-3 normal magic square:

15 is the smallest number that can be factorized using Shor’s quantum algorithm.

There are 15 solutions to Znám’s problem of length 7.

15 is a repdigit in binary (1111). In hexadecimal, as well as all higher bases, fifteen is represented as F.

And finally, and obviously most importantly, fifteen is the number of years that the fabulous Mrs Weir has put up with my wildly ordinary idiosyncrasies. Fair play eh ?