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

Author

Wesley Ivan Hurt, Jan 31 2022

Keywords

• nonn

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

Examples

			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.
		

Links

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

Crossrefs

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

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

Programs

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 *)

PARI

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

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)).

Conjecture: asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} 1/A020492(k) = 2.4343... (assuming empirically that this sum of reciprocals converges). - Amiram Eldar, Dec 27 2024