A351193 Sum of the 5th powers of primes dividing n.

0, 32, 243, 32, 3125, 275, 16807, 32, 243, 3157, 161051, 275, 371293, 16839, 3368, 32, 1419857, 275, 2476099, 3157, 17050, 161083, 6436343, 275, 3125, 371325, 243, 16839, 20511149, 3400, 28629151, 32, 161294, 1419889, 19932, 275, 69343957, 2476131, 371536, 3157

Offset

1,2

Comments

Inverse Möbius transform of n^5 * c(n), where c(n) is the prime characteristic (A010051). - Wesley Ivan Hurt, Jun 22 2024

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{p|n, p prime} p^5.

G.f.: Sum_{k>=1} prime(k)^5 * x^prime(k) / (1 - x^prime(k)). - Ilya Gutkovskiy, Feb 16 2022

Additive with a(p^e) = p^5. - Amiram Eldar, Jun 20 2022

a(n) = Sum_{d|n} d^5 * c(d), where c = A010051. - Wesley Ivan Hurt, Jun 22 2024

Maple

f:= n -> add(p^5, p = numtheory:-factorset(n)):
map(f, [$1..100]); # Robert Israel, Feb 18 2022

Mathematica

Array[DivisorSum[#, #^5 &, PrimeQ] &, 50]
f[p_, e_] := p^5; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 20 2022 *)

Crossrefs

Sum of the k-th powers of the primes dividing n for k=0..10 : A001221 (k=0), A008472 (k=1), A005063 (k=2), A005064 (k=3), A005065 (k=4), this sequence (k=5), A351194 (k=6), A351195 (k=7), A351196 (k=8), A351197 (k=9), A351198 (k=10).

Cf. A010051.

Sequence in context: A118999 A321829 A111450 * A070332 A373291 A111442

Adjacent sequences: A351190 A351191 A351192 * A351194 A351195 A351196

Keyword

nonn

Author

Wesley Ivan Hurt, Feb 04 2022

Status

approved