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Numbers k for which there is an odd number of nonzero digits when k is written in the factorial base (A007623).
9

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

%S 1,2,4,6,9,11,12,15,17,18,21,23,24,27,29,31,32,34,37,38,40,43,44,46,

%T 48,51,53,55,56,58,61,62,64,67,68,70,72,75,77,79,80,82,85,86,88,91,92,

%U 94,96,99,101,103,104,106,109,110,112,115,116,118,120,123,125

%N Numbers k for which there is an odd number of nonzero digits when k is written in the factorial base (A007623).

%C This sequence offers one possible analog to A000069 (odious numbers) in the factorial base system. A227149 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, A227130.

%C The sequence gives the positions of odd permutations in the tables A060117 and A060118.

%H Antti Karttunen, <a href="/A227132/b227132.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, c = 0, r}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, If[r != 0, c++]; m++]; OddQ[c]]; Select[Range[150], q] (* _Amiram Eldar_, Jan 24 2024 *)

%o (Scheme, with _Antti Karttunen_'s IntSeq-library): (define A227132 (MATCHING-POS 1 0 (lambda (i) (odd? (A060130 i)))))

%Y Complement: A227130.

%Y Cf. A007623, A060117, A060118.

%Y Cf. also A000069, A060130, A227149.

%K nonn,base

%O 1,2

%A _Antti Karttunen_, Jul 02 2013