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

1,2

Comments

Row sums of A103994 (conjectured).

Links

Andrew Howroyd, Table of n, a(n) for n = 1..8192

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

Example

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).

Mathematica

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

Prog

(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 */

Crossrefs

Cf. A036987, A103994, A209229.

Sequence in context: A138806 A181105 A142971 * A114511 A085199 A338504

Adjacent sequences: A104114 A104115 A104116 * A104118 A104119 A104120

Keyword

nonn,easy,mult

Author

Gary W. Adamson, Apr 15 2007

Extensions

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

Status

approved