A351112 Number of balanced numbers dividing n.

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

Offset

1,2

Comments

A balanced number k is a number such that phi(k) | sigma(k).

Inverse Möbius transform of the characteristic function of balanced numbers (A351114). - Wesley Ivan Hurt, Jun 23 2024

Links

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

Formula

a(n) = Sum_{d|n, phi(d)|sigma(d)} 1.

a(n) = Sum_{d|n} A351114(d).

a(n) = tau(n) - Sum_{d|n} sign(sigma(d) mod phi(d)).

Example

a(4) = 2; the balanced divisors of 4 are 1 and 2.
a(5) = 1; 1 is the only balanced divisor of 5.
a(6) = 4; the balanced divisors of 6 are 1,2,3,6.

Maple

f:= proc(n) uses numtheory;
  nops(select(t -> sigma(t) mod phi(t) = 0, divisors(n)))
end proc:
map(f, [$1..100]); # Robert Israel, Nov 28 2023

Mathematica

a[n_] := DivisorSum[n, 1 &, Divisible[DivisorSigma[1, #], EulerPhi[#]] &]; Array[a, 100] (* Amiram Eldar, Feb 01 2022 *)

Prog

(PARI) a(n) = sumdiv(n, d, if (!(sigma(d) % eulerphi(d)), 1)); \\ Michel Marcus, Feb 01 2022

Crossrefs

Cf. A351113 (sum of the balanced numbers dividing n).

Cf. A000005 (tau), A000010 (phi), A000203 (sigma), A020492 (balanced numbers), A023897, A351114.

Sequence in context: A289778 A277523 A144393 * A089400 A239209 A180824

Adjacent sequences: A351109 A351110 A351111 * A351113 A351114 A351115

Keyword

nonn

Author

Wesley Ivan Hurt, Jan 31 2022

Status

approved