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

1,2

Comments

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

Links

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

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)).

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

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

Sequence in context: A279676 A279588 A279590 * A201935 A225445 A167877

Adjacent sequences: A351110 A351111 A351112 * A351114 A351115 A351116

Keyword

nonn

Author

Wesley Ivan Hurt, Jan 31 2022

Status

approved