A307314 Number of divisors d of 2n such that adding d to 2n in binary requires no carries.

1, 2, 1, 3, 2, 3, 1, 4, 2, 3, 1, 5, 1, 2, 1, 5, 2, 5, 1, 5, 2, 2, 1, 7, 2, 2, 2, 4, 1, 3, 1, 6, 2, 4, 1, 7, 2, 3, 1, 7, 2, 4, 1, 3, 2, 2, 1, 9, 1, 4, 2, 3, 1, 4, 1, 6, 1, 2, 1, 6, 1, 2, 1, 7, 4, 6, 1, 6, 2, 3, 1, 10, 2, 3, 1, 4, 1, 3, 1, 9

Offset: 1

Author

Charlie Neder, Apr 02 2019

Keywords

• nonn

Comments

Equivalently, number of numbers d such that d|2n and d AND 2n = 0.

First differences of either bisection of A325123.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A325123, A001511, A000005, A247068.

Programs

Mathematica

a[n_] := DivisorSum[2*n, 1 &, BitAnd[#, 2*n] == 0 &]; Array[a, 100] (* Amiram Eldar, Feb 08 2025 *)

PARI

a(n) = sumdiv(2*n, d, bitand(d, 2*n) == 0); \\ Michel Marcus, Apr 02 2019

Formula

A001511(n) <= a(n) <= A000005(n).

a(p) = 1 + [p is in A247068] for p prime, where [] is the Iverson bracket.