A386591 Number of divisors of n that are not balanced numbers.

0, 0, 0, 1, 1, 0, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 4, 2, 2, 1, 3, 2, 2, 2, 3, 1, 2, 1, 4, 2, 2, 2, 4, 1, 2, 2, 6, 1, 2, 1, 4, 3, 2, 1, 5, 2, 4, 2, 4, 1, 4, 3, 4, 2, 2, 1, 5, 1, 2, 4, 5, 3, 4, 1, 4, 2, 3, 1, 7, 1, 2, 3, 4, 3, 3, 1, 8, 3, 2, 1, 5, 3, 2, 2, 6, 1, 6, 3, 4, 2, 2, 3, 7, 1, 3, 4, 7

Offset: 1

Author

Wesley Ivan Hurt, Jul 26 2025

Keywords

• nonn
• changed

Comments

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

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

Links

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

Crossrefs

Cf. A000005 (tau), A000010 (phi), A000203 (sigma), A020492 (balanced numbers), A351112, A351114.

Programs

Maple

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

Mathematica

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

PARI

a(n) = sumdiv(n, d, sigma(d)%eulerphi(d) != 0); \\ Michel Marcus, Aug 26 2025

Formula

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

a(n) = A000005(n) - A351112(n).