A332775 a(n) = n + sopf(n) - omega(n).

1, 3, 5, 5, 9, 9, 13, 9, 11, 15, 21, 15, 25, 21, 21, 17, 33, 21, 37, 25, 29, 33, 45, 27, 29, 39, 29, 35, 57, 37, 61, 33, 45, 51, 45, 39, 73, 57, 53, 45, 81, 51, 85, 55, 51, 69, 93, 51, 55, 55, 69, 65, 105, 57, 69, 63, 77, 87, 117, 67, 121, 93, 71, 65, 81, 79, 133, 85, 93, 81, 141

Offset: 1

Author

Wesley Ivan Hurt, Jun 08 2020

Keywords

• nonn
• easy

Comments

From Bernard Schott, Jun 10 2020: (Start)

All terms are odd, but not all odd integers are obtained: see A353046.

1 <= a(n) <= 2n-1 (see formula). (End)

Crossrefs

Cf. A001221 (omega), A008472 (sopf), A010051, A055631, A353046.

Programs

Mathematica

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

PARI

a(n) = n + vecsum(factor(n)[, 1]) - omega(n); \\ Michel Marcus, Jul 21 2020

Formula

a(n) = Sum_{k=1..n} k^(c(k)*(1 - ceiling(n/k) + floor(n/k))), where c is the prime characteristic (A010051).

a(n) = n + A055631(n).

From Bernard Schott, Jun 10 2020: (Start)

a(n) = 1 iff n = 1.

a(n) = 2*n-1 iff n is prime.

a(p^k) = p^k + p - 1 for p prime, k > 0. (End)