A359079 a(n) is the sum of the divisors d of 2*n such that the binary expansions of d and 2*n have no common 1-bit.

1, 3, 1, 7, 6, 6, 1, 15, 10, 13, 1, 16, 1, 3, 1, 31, 18, 33, 1, 32, 22, 3, 1, 36, 6, 3, 10, 14, 1, 6, 1, 63, 34, 54, 1, 70, 38, 22, 1, 70, 42, 48, 1, 7, 6, 3, 1, 76, 1, 38, 18, 7, 1, 24, 1, 36, 1, 3, 1, 21, 1, 3, 1, 127, 84, 116, 1, 126, 70, 38, 1, 153, 74, 77

Offset: 1

Author

Rémy Sigrist, Dec 15 2022

Keywords

• nonn
• base

Comments

Odd numbers share a 1-bit (2^0) with all their divisors, hence this sequence deals with even numbers.

Examples

			For n = 6:
- the divisors of 12 are:
      d   bin(d)  common bit?
      --  ------  -----------
       1       1  no
       2      10  no
       3      11  no
       4     100  yes
       6     110  yes
      12    1100  yes
- hence a(6) = 1 + 2 + 3 = 6.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to divisors

Crossrefs

Cf. A246601, A346878.

Programs

Mathematica

a[n_] := DivisorSum[2n, #*Boole[BitAnd[#, 2n] == 0] &]; Array[a, 74]

PARI

a(n) = sumdiv(2*n, d, if (bitand(2*n,d)==0, d, 0))

Python

from sympy import divisors as divs
def a(n): return sum(d for d in divs(2*n, generator=True) if (d>>1)&n == 0)
print([a(n) for n in range(1, 75)]) # Michael S. Branicky, Dec 15 2022

Formula

a(n) <= A346878(n) with equality iff n is a power of 2.