A104117 For n=2^k, a(n) = k+1, else 0.

1, 2, 0, 3, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Gary W. Adamson, Apr 15 2007

Keywords

• nonn
• easy
• mult

Comments

Row sums of A103994 (conjectured).

Examples

			a(8) = 4 = sum of row 8 terms of A103994: (1 + 1 + 0 + 1 + 0 + 0 + 0 + 1).
a(8) = 4 = 1 + log_2(8).
		

Crossrefs

Cf. A036987, A103994, A209229.

Programs

Mathematica

a[n_] := Module[{e = IntegerExponent[n, 2]}, If[n == 2^e, e+1, 0]]; Array[a, 100] (* Amiram Eldar, Aug 31 2023 *)

PARI

a(n)=direuler(p=1,n,if(p==2,1/(1-X)^2,1))[n] /* Ralf Stephan, Mar 28 2015 */

PARI

a(n)=if(n==2^valuation(n,2),valuation(n,2)+1,0) /* Ralf Stephan, Mar 28 2015 */

Formula

a(n) = 1 + log_2(n), for n = 1, 2, 4, 8, ... and the rest zeros.

Dirichlet g.f.: 1/(1-2^(-s))^2, i.e., Dirichlet convolution of A036987 (right-shifted, assuming offset 1 there) with itself.

Multiplicative with a(2^e) = 1+e, and a(p^e) = 0 for odd primes p and e>=1. Dirichlet convolution square of A209229. - R. J. Mathar, Mar 12 2012

Extensions

More terms and better name from Ralf Stephan, Mar 28 2015