A351195 Sum of the 7th powers of the primes dividing n.

0, 128, 2187, 128, 78125, 2315, 823543, 128, 2187, 78253, 19487171, 2315, 62748517, 823671, 80312, 128, 410338673, 2315, 893871739, 78253, 825730, 19487299, 3404825447, 2315, 78125, 62748645, 2187, 823671, 17249876309, 80440, 27512614111, 128, 19489358, 410338801

Offset

1,2

Comments

Inverse Möbius transform of n^7 * c(n), where c(n) is the prime characteristic (A010051). - Wesley Ivan Hurt, Jun 22 2024

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{p|n, p prime} p^7.

G.f.: Sum_{k>=1} prime(k)^7 * x^prime(k) / (1 - x^prime(k)). - Ilya Gutkovskiy, Feb 16 2022

Additive with a(p^e) = p^7. - Amiram Eldar, Jun 20 2022

a(n) = Sum_{d|n} d^7 * c(d), where c = A010051. - Wesley Ivan Hurt, Jun 22 2024

Mathematica

Array[DivisorSum[#, #^7 &, PrimeQ] &, 50]
f[p_, e_] := p^7; 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), A351193 (k=5), A351194 (k=6), this sequence (k=7), A351196 (k=8), A351197 (k=9), A351198 (k=10).

Cf. A010051.

Sequence in context: A128696 A195594 A321831 * A223954 A224138 A331198

Adjacent sequences: A351192 A351193 A351194 * A351196 A351197 A351198

Keyword

nonn

Author

Wesley Ivan Hurt, Feb 04 2022

Status

approved