A386592 Sum of the divisors of n that are not balanced numbers.

0, 0, 0, 4, 5, 0, 7, 12, 9, 15, 11, 4, 13, 7, 5, 28, 17, 27, 19, 39, 28, 33, 23, 36, 30, 39, 36, 39, 29, 15, 31, 60, 44, 51, 12, 67, 37, 57, 52, 87, 41, 28, 43, 81, 59, 69, 47, 100, 56, 90, 68, 95, 53, 108, 71, 47, 76, 87, 59, 99, 61, 93, 100, 124, 83, 132, 67, 123, 92, 22, 71, 171, 73, 111, 105, 137, 95, 78, 79, 183, 117, 123, 83, 144, 107, 129, 116, 177

Offset: 1

Author

Wesley Ivan Hurt, Jul 26 2025

Keywords

• nonn
• look
• changed

Comments

Sum of the divisors d of n such that phi(d) does not divide sigma(d).

Inverse Möbius transform of n * (1 - c(n)), where c = A351114.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010 (phi), A000203 (sigma), A351113, A351114, A387333.

Programs

Maple

g:= proc(n) option remember; numtheory:-sigma(n) mod numtheory:-phi(n) <> 0 end proc:
f:= n -> convert(select(g,numtheory:-divisors(n)),`+`):
map(f, [$1..100]); # Robert Israel, Aug 26 2025

Mathematica

Table[Sum[d (Ceiling[DivisorSigma[1, d]/EulerPhi[d]] - Floor[DivisorSigma[1, d]/EulerPhi[d]]), {d, Divisors[n]}], {n, 100}]

Formula

a(n) = Sum_{d|n} d * (1 - c(d)), where c = A351114.

a(n) = A000203(n) - A351113(n).