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

0,5

Links

Antoine Mathys, Table of n, a(n) for n = 0..9999

Project Euler, Problem 704 - Factors of Two in Binomial Coefficients.

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).

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 *)

Prog

(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;

Crossrefs

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

Sequence in context: A338281 A338723 A175708 * A023830 A063649 A053158

Adjacent sequences: A373706 A373707 A373708 * A373710 A373712 A373713

Keyword

nonn,base,easy

Author

Antoine Mathys, Jun 14 2024

Status

approved