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

1,8

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

Formula

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

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

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}]

Prog

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

Crossrefs

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

Sequence in context: A378375 A156144 A358235 * A136044 A375736 A184240

Adjacent sequences: A386588 A386589 A386590 * A386592 A386593 A386594

Keyword

nonn

Author

Wesley Ivan Hurt, Jul 26 2025

Status

approved