A322078 -id:A322078 - cp's OEIS frontend

A382511 Expansion of Sum_{p prime} x^p / (1 - x^p)^3.

0, 1, 1, 3, 1, 9, 1, 10, 6, 18, 1, 31, 1, 31, 21, 36, 1, 66, 1, 65, 34, 69, 1, 114, 15, 94, 45, 115, 1, 196, 1, 136, 72, 156, 43, 249, 1, 193, 97, 246, 1, 357, 1, 263, 165, 279, 1, 436, 28, 380, 159, 361, 1, 549, 81, 442, 196, 438, 1, 753, 1, 499, 276, 528, 106

Offset: 1

Ilya Gutkovskiy, Mar 30 2025

nonn

Cf. A000217, A001221, A007437, A069359, A305614, A322078.

nmax = 65; CoefficientList[Series[Sum[x^Prime[k]/(1 - x^Prime[k])^3, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = Sum_{p|n, p prime} A000217(n/p).

a(n) = (A069359(n) + A322078(n)) / 2.

A382512 Expansion of Sum_{p prime} x^p / (1 - x^p)^p.

Original entry on oeis.org

0, 1, 1, 2, 1, 6, 1, 4, 6, 10, 1, 16, 1, 14, 30, 8, 1, 30, 1, 45, 56, 22, 1, 48, 70, 26, 45, 98, 1, 196, 1, 16, 132, 34, 420, 96, 1, 38, 182, 350, 1, 588, 1, 308, 615, 46, 1, 160, 924, 740, 306, 481, 1, 198, 2002, 1744, 380, 58, 1, 1605, 1, 62, 3234, 32, 3640

Offset: 1

Ilya Gutkovskiy, Mar 30 2025

nonn

Cf. A001221, A069359, A157019, A322078, A373458, A373459.

nmax = 65; CoefficientList[Series[Sum[x^Prime[k]/(1 - x^Prime[k])^Prime[k], {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = Sum_{p|n, p prime} binomial(n/p+p-2, p-1).

cp's OEIS Frontend

A382511 Expansion of Sum_{p prime} x^p / (1 - x^p)^3.

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