A227148 Numbers k for which the sum of digits is even when k is written in the factorial base (A007623).

0, 3, 4, 7, 8, 11, 12, 15, 16, 19, 20, 23, 25, 26, 29, 30, 33, 34, 37, 38, 41, 42, 45, 46, 48, 51, 52, 55, 56, 59, 60, 63, 64, 67, 68, 71, 73, 74, 77, 78, 81, 82, 85, 86, 89, 90, 93, 94, 96, 99, 100, 103, 104, 107, 108, 111, 112, 115, 116, 119, 121, 122, 125

Offset: 1

Author

Antti Karttunen, Jul 02 2013

Keywords

• nonn
• base

Comments

Numbers k for which minimal number of factorials needed to add to get k is even.

This sequence offers one possible analog to A001969 (evil numbers) in factorial base system. A227130 gives another kind of analog.

In each range [0,n!-1] exactly half of the integers are found in this sequence, and the other half of them are found in the complement, A227149.

The sequence gives the positions of even permutations in the tables A055089 and A195663; and equivalently, the positions of even numbers in A055091.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..2520
• Wikipedia, Parity of permutation.

Crossrefs

Complement: A227149. Cf. also A001969, A034968, A227130.

Programs

Mathematica

q[n_] := Module[{k = n, m = 2, s = 0, r}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, s += r; m++]; EvenQ[s]]; Select[Range[0, 125], q] (* Amiram Eldar, Jan 24 2024 *)