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September 30[edit]

Exploring subfactorials[edit]

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]