A386438 a(n) = sigma(n) + omega(n) - n * Sum_{p|n, p prime} 1 / p.

1, 3, 4, 6, 6, 9, 8, 12, 11, 13, 12, 20, 14, 17, 18, 24, 18, 26, 20, 30, 24, 25, 24, 42, 27, 29, 32, 40, 30, 44, 32, 48, 36, 37, 38, 63, 38, 41, 42, 64, 42, 58, 44, 60, 56, 49, 48, 86, 51, 60, 54, 70, 54, 77, 58, 86, 60, 61, 60, 109, 62, 65, 76, 96, 68, 86, 68, 90, 72, 88, 72, 137, 74, 77, 86, 100, 80, 100, 80, 132, 95, 85, 84, 145, 88, 89, 90, 130, 90, 144

Offset: 1

Author

Wesley Ivan Hurt, Jul 21 2025

Keywords

• nonn

Comments

For each divisor d of n, add 1 if n/d is prime, else add d.

Crossrefs

Cf. A000010 (phi), A000203 (sigma), A001221 (omega), A005171, A007503, A010051, A069359, A348219.

Programs

Mathematica

Table[Sum[d^(1 - PrimePi[n/d] + PrimePi[n/d - 1]), {d, Divisors[n]}], {n, 100}]

Formula

a(n) = Sum_{d|n} d^c(n/d), where c = A005171.

a(n) = Sum_{d|n} (d + c(d) - phi(d)*omega(n/d)), where c = A010051.

a(n) = A000203(n) + A001221(n) - A069359(n).

a(n) = A007503(n) - A348219(n).