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A349217
a(n) = Sum_{d|n} n^c(d), where c is the prime characteristic (A010051).
0
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.
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
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Nov 10 2021
STATUS
approved