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

1,2

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

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

Mathematica

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

Prog

(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

Crossrefs

Inverse Moebius transform of A089026.

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

Sequence in context: A317645 A117571 A008474 * A111611 A100478 A112376

Adjacent sequences: A349212 A349213 A349214 * A349216 A349217 A349218

Keyword

nonn

Author

Wesley Ivan Hurt, Nov 10 2021

Status

approved