A351113 Sum of the balanced numbers dividing n.

1, 3, 4, 3, 1, 12, 1, 3, 4, 3, 1, 24, 1, 17, 19, 3, 1, 12, 1, 3, 4, 3, 1, 24, 1, 3, 4, 17, 1, 57, 1, 3, 4, 3, 36, 24, 1, 3, 4, 3, 1, 68, 1, 3, 19, 3, 1, 24, 1, 3, 4, 3, 1, 12, 1, 73, 4, 3, 1, 69, 1, 3, 4, 3, 1, 12, 1, 3, 4, 122, 1, 24, 1, 3, 19, 3, 1, 90, 1, 3, 4, 3, 1, 80

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).

Examples

			a(4) = 3; the balanced divisors of 4 are 1 and 2 and 1+2 = 3.
a(5) = 1; 1 is the only balanced divisor of 5.
a(6) = 12; the balanced divisors of 6 are 1,2,3,6 and 1+2+3+6 = 12.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A351112 (number of balanced divisors of n).

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

Programs

Mathematica

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

PARI

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

Formula

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

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

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