A386589 a(n) = Sum_{d|n} d^c(d), where c = A351114.

1, 3, 4, 4, 2, 12, 2, 5, 5, 5, 2, 25, 2, 18, 20, 6, 2, 14, 2, 7, 6, 5, 2, 27, 3, 5, 6, 20, 2, 59, 2, 7, 6, 5, 38, 28, 2, 5, 6, 9, 2, 70, 2, 7, 22, 5, 2, 29, 3, 7, 6, 7, 2, 16, 4, 77, 6, 5, 2, 74, 2, 5, 8, 8, 4, 16, 2, 7, 6, 125, 2, 31, 2, 5, 22, 7, 4, 93, 2, 11, 7, 5, 2, 85, 4, 5, 6, 9, 2, 63, 4, 7, 6, 5, 4, 31, 2, 20, 8, 10

Offset: 1

Author

Wesley Ivan Hurt, Jul 26 2025

Keywords

• nonn

Comments

Inverse Möbius transform of n^c(n), where c = A351114.

For each divisor d of n, add d if d is a balanced number (A020492), else add 1.

Crossrefs

Cf. A000005 (tau), A020492 (balanced numbers), A351112, A351113, A351114.

Programs

Mathematica

Table[Sum[d^(1 - Ceiling[DivisorSigma[1, d]/EulerPhi[d]] + Floor[DivisorSigma[1, d]/EulerPhi[d]]), {d, Divisors[n]}], {n, 100}]

Formula

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

a(n) = A000005(n) + A351113(n) - A351112(n).