Mathematics desk
< September 29	<< Aug \| September \| Oct >>	Current desk >

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

September 30

Exploring subfactorials

The subfactorials are the numbers of ways to arrange the numbers 1 to n in a group of slots numbered 1 through n so that no number is in the slot with that number. The subfactorials of the integers 1 through 10 are 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, and 1334961. I think it's good to know a few properties subfactorials must have:

All odd subfactorials are one more than a multiple of 8.
No even subfactorial is one more than a multiple of 3.
No odd subfactorial is one less than a multiple of 3.
No subfactorial ends in 7 or 8.

Any other known properties?? Georgia guy (talk) 15:38, 30 September 2022 (UTC)[reply]

See oeis:A000166. —Kusma (talk) 15:54, 30 September 2022 (UTC)[reply]

Search there for "periodic". This means that the sequence of remainders when dividing by given

m

repeats, with each repeat marked by the (re)appearance of the remainders 1 – 0 – 1. So you have to examine only a finite amount. That way I saw that the statements above for multiples of 3 apply without further change also for multiples of 9 (but not for multiples of 27). Another property: no subfactorial differs by more than 2 from a multiple of 11. Are these properties interesting? Unless there is a way to connect this to other parts of number theory, do not expect much excitement. --Lambiam 21:01, 30 September 2022 (UTC)[reply]