A305426 Number of proper divisors of n of the form 2^k - 1 for k >= 1.

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

Offset: 1

Author

Antti Karttunen, Jun 11 2018

Keywords

• nonn

Comments

a(n) is the number of terms of A000225 less than n that divide n.

Links

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

Crossrefs

Cf. A000225, A036987, A154402.

Cf. also A305435.

Programs

Mathematica

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

PARI

A209229(n) = (n && !bitand(n,n-1));
A036987(n) = A209229(1+n);
A305426(n) = sumdiv(n,d,(dA036987(d));

Formula

a(n) = Sum_{d|n, dA036987(d).

a(n) = A154402(n) - A036987(n).