A348915 a(n) = Sum_{d|n} d^(d mod 2).

1, 2, 4, 3, 6, 6, 8, 4, 13, 8, 12, 8, 14, 10, 24, 5, 18, 16, 20, 10, 32, 14, 24, 10, 31, 16, 40, 12, 30, 28, 32, 6, 48, 20, 48, 19, 38, 22, 56, 12, 42, 36, 44, 16, 78, 26, 48, 12, 57, 34, 72, 18, 54, 44, 72, 14, 80, 32, 60, 32, 62, 34, 104, 7, 84, 52, 68, 22, 96, 52, 72, 22, 74

Offset: 1

Author

Wesley Ivan Hurt, Nov 03 2021

Keywords

• nonn

Comments

For each divisor d of n, add d if d is odd, otherwise add 1.

Inverse Möbius transform of n^(n mod 2). - Wesley Ivan Hurt, Mar 31 2025

Examples

			For n = 12, the divisors of 12 are 1, 2, 3, 4, 6, 12 with corresponding summands 1, 1, 3, 1, 1, 1, respectively. The sum is then a(12) = 1 + 1 + 3 + 1 + 1 + 1 = 8.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000005 (tau), A000203 (sigma), A000593, A065608, A183063.

Programs

Maple

f:= proc(n) local d;
  add(d^(d mod 2), d = numtheory:-divisors(n))
end proc;
map(f, [$1..100]); # Robert Israel, Jun 01 2025

Mathematica

a[n_] := DivisorSum[n, #^Mod[#, 2] &]; Array[a, 100] (* Amiram Eldar, Nov 04 2021 *)

PARI

a(n) = sumdiv(n, d, if (d%2, d, 1)); \\ Michel Marcus, Nov 04 2021

Python

from math import prod
from sympy import factorint
def A348915(n):
    f = factorint(n>>(m:=(~n&n-1).bit_length())).items()
    d = prod(e+1 for p,e in f)
    s = prod((p**(e+1)-1)//(p-1) for p, e in f)
    return s+d*m # Chai Wah Wu, Jul 16 2022

Formula

a(n) = A000593(n) + A183063(n).

a(n) = A065608(2n) - 2*A065608(n).

a(p) = p+1 for odd primes p. - Wesley Ivan Hurt, Nov 28 2021

a(n) = A000203(A000265(n))+A000005(A000265(n))*A007814(n). - Chai Wah Wu, Jul 16 2022

a(n) = A000203(n) if n is odd. - Robert Israel, Jun 01 2025