A373709 Partial sums of A119387.

0, 0, 1, 1, 3, 4, 6, 6, 9, 11, 14, 15, 18, 20, 23, 23, 27, 30, 34, 36, 40, 43, 47, 48, 52, 55, 59, 61, 65, 68, 72, 72, 77, 81, 86, 89, 94, 98, 103, 105, 110, 114, 119, 122, 127, 131, 136, 137, 142, 146, 151, 154, 159, 163, 168, 170, 175, 179, 184, 187, 192

Offset: 0

Author

Antoine Mathys, Jun 14 2024

Keywords

• nonn
• base
• easy

Links

• Antoine Mathys, Table of n, a(n) for n = 0..9999
• Project Euler, Problem 704 - Factors of Two in Binomial Coefficients.

Crossrefs

Cf. A000120, A001855, A005187, A070939, A119387.

Programs

Maple

a:= proc(n) option remember; `if`(n<0, 0,
      a(n-1)+ilog2(n+1)-padic[ordp](n+1, 2))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 23 2024

Mathematica

Accumulate[Table[BitLength[k] - 1 - IntegerExponent[k, 2], {k, 100}]] (* Paolo Xausa, Oct 01 2024 *)

PARI

bit_width(n)=logint(n,2)+1;
a(n)=my(d=bit_width(n+1),p=hammingweight(n+1));(n+2)*d-2*n-2^d+p-1;

Formula

a(n) = Sum_{m = 0..n} A119387(m).

a(n) = (n+2)*d - 2*n - 2^d + p - 1, with d = bit_width(n+1) = A070939(n+1) and p = popcount(n+1) = A000120(n+1).

a(n) = A001855(n+2) - A005187(n+1).