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

1,2

Comments

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

Links

Table of n, a(n) for n=1..73.

Index entries for sequences related to sigma(n)

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A309131 A162953 A235451 * A039819 A242424 A232271

Adjacent sequences: A348912 A348913 A348914 * A348916 A348917 A348918

Keyword

nonn

Author

Wesley Ivan Hurt, Nov 03 2021

Status

approved