A349215 a(n) = Sum_{d|n} d^c(d), where c is the prime characteristic (A010051).

1, 3, 4, 4, 6, 7, 8, 5, 5, 9, 12, 9, 14, 11, 10, 6, 18, 9, 20, 11, 12, 15, 24, 11, 7, 17, 6, 13, 30, 15, 32, 7, 16, 21, 14, 12, 38, 23, 18, 13, 42, 17, 44, 17, 12, 27, 48, 13, 9, 11, 22, 19, 54, 11, 18, 15, 24, 33, 60, 19, 62, 35, 14, 8, 20, 21, 68, 23, 28, 19, 72, 15, 74, 41, 12

Offset: 1

Author

Wesley Ivan Hurt, Nov 10 2021

Keywords

• nonn

Comments

For each divisor d of n, add d if d is prime, otherwise add 1. For example, a(6) = 1 + 2 + 3 + 1 = 7.

Links

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

Crossrefs

Inverse Moebius transform of A089026.

Cf. A010051, A000005 (tau), A008472 (sopf), A001221 (omega).

Programs

Mathematica

a[n_] := DivisorSum[n, #^Boole[PrimeQ[#]] &]; Array[a, 75] (* Amiram Eldar, Nov 11 2021 *)

PARI

a(n) = sumdiv(n, d, if (isprime(d), d, 1)); \\ Michel Marcus, Nov 11 2021

Python

from math import prod
from sympy import factorint
def A349215(n):
    fs = factorint(n)
    return sum(a-1 for a in fs.keys())+prod(1+d for d in fs.values()) # Chai Wah Wu, Nov 11 2021

Formula

a(n) = A000005(n)+A008472(n)-A001221(n). - Chai Wah Wu, Nov 11 2021

a(p) = p+1 for primes p. - Wesley Ivan Hurt, Nov 28 2021