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Numbers k for which the sum of digits is even when k is written in the factorial base (A007623).
8

%I #20 Jan 24 2024 01:49:08

%S 0,3,4,7,8,11,12,15,16,19,20,23,25,26,29,30,33,34,37,38,41,42,45,46,

%T 48,51,52,55,56,59,60,63,64,67,68,71,73,74,77,78,81,82,85,86,89,90,93,

%U 94,96,99,100,103,104,107,108,111,112,115,116,119,121,122,125

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

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

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

%C 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.

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

%H Antti Karttunen, <a href="/A227148/b227148.txt">Table of n, a(n) for n = 1..2520</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Parity_of_a_permutation">Parity of permutation</a>.

%t 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 *)

%o (Scheme, with _Antti Karttunen_'s IntSeq-library): (define A227148 (MATCHING-POS 1 0 (lambda (i) (even? (A034968 i)))))

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

%K nonn,base

%O 1,2

%A _Antti Karttunen_, Jul 02 2013