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.

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, 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, 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

Offset: 0

Author

Antti Karttunen, Aug 09 2016

Keywords

• nonn
• base

Comments

Each natural numbers occurs an infinite number of times.

Can be used when computing A275727.

Examples

			22 has factorial base representation "320" (= A007623(22)), which does not contain any "1". Thus a(22) = 0, as the empty sum is 0.
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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..40320
• Index entries for sequences related to binary expansion of n
• Index entries for sequences related to factorial base representation

Crossrefs

Cf. A000079, A000120, A000225, A007489, A048675, A257261, A257511, A275727, A275730.

Left inverse of A059590.

Cf. A255411 (indices of zeros).

Cf. also A275732.

Programs

Mathematica

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 *)

Formula

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].

Other identities and observations. For all n >= 0:

a(n) = A048675(A275732(n)).

A000120(a(n)) = A257511(n).

a(A007489(n)) = A000225(n).

a(A059590(n)) = n.

a(A255411(n)) = 0.