A351194 Sum of the 6th powers of the primes dividing n.

0, 64, 729, 64, 15625, 793, 117649, 64, 729, 15689, 1771561, 793, 4826809, 117713, 16354, 64, 24137569, 793, 47045881, 15689, 118378, 1771625, 148035889, 793, 15625, 4826873, 729, 117713, 594823321, 16418, 887503681, 64, 1772290, 24137633, 133274, 793, 2565726409

Offset

1,2

Comments

Inverse Möbius transform of n^6 * 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^6.

G.f.: Sum_{k>=1} prime(k)^6 * x^prime(k) / (1 - x^prime(k)). - Ilya Gutkovskiy, Feb 16 2022

Additive with a(p^e) = p^6. - Amiram Eldar, Jun 20 2022

a(n) = Sum_{d|n} d^6 * c(d), where c = A010051. - Wesley Ivan Hurt, Jun 22 2024

Mathematica

Array[DivisorSum[#, #^6 &, PrimeQ] &, 50]
f[p_, e_] := p^6; 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), this sequence (k=6), A351195 (k=7), A351196 (k=8), A351197 (k=9), A351198 (k=10).

Cf. A010051.

Sequence in context: A321830 A182673 A182674 * A048392 A274219 A161861

Adjacent sequences: A351191 A351192 A351193 * A351195 A351196 A351197

Keyword

nonn

Author

Wesley Ivan Hurt, Feb 04 2022

Status

approved