A380441 Sum of the nonprimes dividing n and the number of distinct primes dividing n.

1, 2, 2, 6, 2, 9, 2, 14, 11, 13, 2, 25, 2, 17, 18, 30, 2, 36, 2, 37, 24, 25, 2, 57, 27, 29, 38, 49, 2, 65, 2, 62, 36, 37, 38, 88, 2, 41, 42, 85, 2, 87, 2, 73, 72, 49, 2, 121, 51, 88, 54, 85, 2, 117, 58, 113, 60, 61, 2, 161, 2, 65, 96, 126, 68, 131, 2, 109, 72, 133, 2, 192, 2, 77, 118, 121, 80, 153, 2, 181, 119, 85, 2, 215, 88, 89, 90, 169, 2, 227, 94, 145, 96

Offset: 1

Author

Wesley Ivan Hurt, Jun 21 2025

Keywords

• nonn
• easy
• changed

Comments

Inverse Möbius transform of A005451(n).

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

Crossrefs

Cf. A000203 (sigma), A001221 (omega), A005171 (char nonprimes), A005451, A008472 (sopf), A023890.

Programs

Mathematica

Table[DivisorSigma[1, n] - Sum[p, {p, Select[Divisors[n], PrimeQ]}] + PrimeNu[n], {n, 100}]

Formula

a(n) = sigma(n) - sopf(n) + omega(n).

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

a(n) = Sum_{d|n} A005451(d).

a(p^k) = 1 - p + (p^(k+1)-1)/(p-1) for p prime, k >= 1. - Wesley Ivan Hurt, Jul 02 2025

a(n) = A023890(n) + A001221(n). - Wesley Ivan Hurt, Aug 31 2025