A349217 a(n) = Sum_{d|n} n^c(d), where c is the prime characteristic (A010051).

1, 3, 4, 6, 6, 14, 8, 11, 11, 22, 12, 28, 14, 30, 32, 20, 18, 40, 20, 44, 44, 46, 24, 54, 27, 54, 30, 60, 30, 95, 32, 37, 68, 70, 72, 79, 38, 78, 80, 86, 42, 131, 44, 92, 94, 94, 48, 104, 51, 104, 104, 108, 54, 114, 112, 118, 116, 118, 60, 189, 62, 126, 130, 70, 132, 203, 68, 140

Offset

1,2

Comments

For each divisor d of n, add n if d is prime, otherwise add 1. For example, the divisors of 6 are 1,2,3,6. Then we have 1 + 6 + 6 + 1 = 14.

Links

Table of n, a(n) for n=1..68.

Formula

a(n) = (n-1)*omega(n) + tau(n) = (n-1)*A001221(n) + A000005(n). - Alois P. Heinz, Nov 10 2021

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

Maple

a:= n-> (n-1)*nops(ifactors(n)[2]) + numtheory[tau](n):
seq(a(n), n=1..60);  # Alois P. Heinz, Nov 10 2021

Mathematica

a[n_] := DivisorSum[n, n^Boole[PrimeQ[#]] &]; Array[a, 70] (* Amiram Eldar, Nov 11 2021 *)

Prog

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

Crossrefs

Cf. A000005 (tau), A000040, A001221 (omega), A010051.

Sequence in context: A133689 A285895 A220345 * A065967 A345209 A117986

Adjacent sequences: A349214 A349215 A349216 * A349218 A349219 A349220

Keyword

nonn

Author

Wesley Ivan Hurt, Nov 10 2021

Status

approved