A305436 Number of divisors of n of the form 2^k + 1 for k >= 0.

0, 1, 1, 1, 1, 2, 0, 1, 2, 2, 0, 2, 0, 1, 2, 1, 1, 3, 0, 2, 1, 1, 0, 2, 1, 1, 2, 1, 0, 3, 0, 1, 2, 2, 1, 3, 0, 1, 1, 2, 0, 2, 0, 1, 3, 1, 0, 2, 0, 2, 2, 1, 0, 3, 1, 1, 1, 1, 0, 3, 0, 1, 2, 1, 2, 3, 0, 2, 1, 2, 0, 3, 0, 1, 2, 1, 0, 2, 0, 2, 2, 1, 0, 2, 2, 1, 1, 1, 0, 4, 0, 1, 1, 1, 1, 2, 0, 1, 3, 2, 0, 3, 0, 1, 2

Offset: 1

Author

Antti Karttunen, Jun 11 2018

Keywords

• nonn

Comments

a(n) is the number of terms of A000051 that divide n.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A000051, A209229, A292315 (positions of zeros), A305435, A323482.

Cf. also A154402.

Programs

Mathematica

Table[DivisorSum[n, 1 &, IntegerQ@ Log2[# - 1] &], {n, 105}] (* Michael De Vlieger, Jun 11 2018 *)

PARI

A209229(n) = (n && !bitand(n,n-1));
A305436(n) = sumdiv(n,d,A209229(d-1));

Formula

a(n) = Sum_{d|n} A209229(d-1).

a(n) = A305435(n) + A209229(n-1).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A323482 = 1.264499... . - Amiram Eldar, Dec 31 2023