A348219 a(n) = tau(n) - omega(n) + n * Sum_{p|n, p prime} 1/p.

1, 2, 2, 4, 2, 7, 2, 7, 5, 9, 2, 14, 2, 11, 10, 12, 2, 19, 2, 18, 12, 15, 2, 26, 7, 17, 12, 22, 2, 36, 2, 21, 16, 21, 14, 37, 2, 23, 18, 34, 2, 46, 2, 30, 28, 27, 2, 48, 9, 39, 22, 34, 2, 51, 18, 42, 24, 33, 2, 71, 2, 35, 34, 38, 20, 66, 2, 42, 28, 64, 2, 70, 2, 41, 44, 46

Offset: 1

Author

Wesley Ivan Hurt, Oct 07 2021

Keywords

• nonn

Comments

For each divisor d of n, add n/d if d is prime, otherwise add 1. For example, a(9) = 5 can be found using its divisors 1,3,9 to get 1 + 9/3 + 1 = 5.

If p is prime, then a(p) = 2 since we have a(p) = tau(p) - omega(p) + p/p = 2 - 1 + 1 = 2.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000005 (tau), A000010 (phi), A001221 (omega), A005171, A007503, A010051, A069359, A348203, A386438.

Programs

PARI

A348219(n) = sumdiv(n,d,(n/d)^isprime(d)); \\ Antti Karttunen, Nov 11 2021

PARI

A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A348219(n) = (A069359(n)+numdiv(n)-omega(n)); \\ Antti Karttunen, Nov 11 2021

Formula

a(n) = Sum_{d|n} (n/d)^c(d), where c is the prime characteristic (A010051).

a(n) = A000005(n) - A001221(n) + A069359(n).

a(prime(n)) = 2.

From Wesley Ivan Hurt, Jul 21 2025: (Start)

a(n) = Sum_{d|n} (c(d) + phi(d)*omega(n/d)), where c = A005171.

a(n) = A007503(n) - A386438(n). (End)