A348203 a(n) = n - omega(n) + n * Sum_{p|n} 1/p.

1, 2, 3, 5, 5, 9, 7, 11, 11, 15, 11, 20, 13, 21, 21, 23, 17, 31, 19, 32, 29, 33, 23, 42, 29, 39, 35, 44, 29, 58, 31, 47, 45, 51, 45, 64, 37, 57, 53, 66, 41, 80, 43, 68, 67, 69, 47, 86, 55, 83, 69, 80, 53, 97, 69, 90, 77, 87, 59, 119, 61, 93, 91, 95, 81, 124, 67, 104, 93, 126

Offset: 1

Author

Wesley Ivan Hurt, Oct 06 2021

Keywords

• nonn

Comments

For 1 <= k <= n, if k is a prime divisor of n then add n/k, otherwise add 1. For example, a(6) = 9 since the values of k from 1 to 6 would be: 1 + 6/2 + 6/3 + 1 + 1 + 1 = 9.

If p is prime, then a(p) = p since we have a(p) = p - omega(p) + phi(1)*omega(p/1) + phi(p)*omega(p/p) = p - 1 + 1*1 + (p-1)*0 = p.

Crossrefs

Cf. A000010 (phi), A001221 (omega), A010051, A069359.

Programs

Mathematica

Table[n - PrimeNu[n] + Sum[EulerPhi[k]*PrimeNu[n/k] (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 80}]

Formula

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

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

a(prime(n)) = prime(n).