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A275736 a(n) has base-2 representation with ones in those digit-positions where n contains ones in its factorial base representation, and zeros in all the other positions.

%I A275736 #18 Aug 11 2016 23:21:49
%S A275736 0,1,2,3,0,1,4,5,6,7,4,5,0,1,2,3,0,1,0,1,2,3,0,1,8,9,10,11,8,9,12,13,
%T A275736 14,15,12,13,8,9,10,11,8,9,8,9,10,11,8,9,0,1,2,3,0,1,4,5,6,7,4,5,0,1,
%U A275736 2,3,0,1,0,1,2,3,0,1,0,1,2,3,0,1,4,5,6,7,4,5,0,1,2,3,0,1,0,1,2,3,0,1,0,1,2,3,0,1,4,5,6,7,4,5,0
%N A275736 a(n) has base-2 representation with ones in those digit-positions where n contains ones in its factorial base representation, and zeros in all the other positions.
%C A275736 Each natural numbers occurs an infinite number of times.
%C A275736 Can be used when computing A275727.
%H A275736 Antti Karttunen, <a href="/A275736/b275736.txt">Table of n, a(n) for n = 0..40320</a>
%H A275736 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A275736 <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>
%F A275736 If A257261(n) = 0, then a(n) = 0, otherwise a(n) = A000079(A257261(n)-1) + a(A275730(n, A257261(n)-1)).  [Here A275730(n,p) is a bivariate function that "clears" the digit at zero-based position p in the factorial base representation of n].
%F A275736 Other identities and observations. For all n >= 0:
%F A275736 a(n) = A048675(A275732(n)).
%F A275736 A000120(a(n)) = A257511(n).
%F A275736 a(A007489(n)) = A000225(n).
%F A275736 a(A059590(n)) = n.
%F A275736 a(A255411(n)) = 0.
%e A275736 22 has factorial base representation "320" (= A007623(22)), which does not contain any "1". Thus a(22) = 0, as the empty sum is 0.
%e A275736 35 has factorial base representation "1121" (= A007623(35)). Here 1's occur in the following positions, when counted from right (starting with 0 for the least significant position): 0, 2 and 3. Thus a(35) = 2^0 + 2^2 + 2^3 = 1*4*8 = 13.
%t A275736 nn = 120; m = 1; While[Factorial@ m < nn, m++]; m; Map[FromDigits[#, 2] &[IntegerDigits[#, MixedRadix[Reverse@ Range[2, m]]] /. k_ /; k != 1 -> 0] &, Range[0, nn]] (* _Michael De Vlieger_, Aug 11 2016, Version 10.2 *)
%o A275736 (Scheme, with memoization-macro definec)
%o A275736 (definec (A275736 n) (cond ((zero? (A257261 n)) 0) (else (+ (A000079 (+ -1 (A257261 n))) (A275736 (A275730bi n (- (A257261 n) 1))))))) ;; Code for A275730bi given in A275730.
%Y A275736 Cf. A000079, A000120, A000225, A007489, A048675, A257261, A257511, A275727, A275730.
%Y A275736 Left inverse of A059590.
%Y A275736 Cf. A255411 (indices of zeros).
%Y A275736 Cf. also A275732.
%K A275736 nonn,base
%O A275736 0,3
%A A275736 _Antti Karttunen_, Aug 09 2016