A373459 -id:A373459 - cp's OEIS frontend

A373458 Expansion of Sum_{p prime} x^p/(1 - p*x^p).

0, 1, 1, 2, 1, 7, 1, 8, 9, 21, 1, 59, 1, 71, 106, 128, 1, 499, 1, 637, 778, 1035, 1, 4235, 625, 4109, 6561, 8535, 1, 39192, 1, 32768, 59170, 65553, 18026, 308219, 1, 262163, 531610, 602413, 1, 2659706, 1, 2098483, 5173594, 4194327, 1, 22737515, 117649, 18730341

Offset: 1

Seiichi Manyama, Jun 06 2024

Cf. A001221, A305614, A373459.

A373458 := proc(n)
    local a,d ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if isprime(d) then
            a := a+d^(n/d-1) ;
        end if;
    end do:
    a ;
end proc:
seq(A373458(n),n=1..20) ; # R. J. Mathar, Jun 07 2024

a[n_]:=Sum[Boole[PrimeQ[d]]d^(n/d-1),{d,Divisors[n]}]; Array[a,50] (* Stefano Spezia, Mar 30 2025 *)

a(n) = sumdiv(n, d, isprime(d)*d^(n/d-1));

a(n) = Sum_{p|n prime} p^(n/p - 1).

If p is prime, a(p) = 1.

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).

A382513 Expansion of Sum_{p prime} p * x^p / (1 - p * x^p).

Original entry on oeis.org

0, 2, 3, 4, 5, 17, 7, 16, 27, 57, 11, 145, 13, 177, 368, 256, 17, 1241, 19, 1649, 2530, 2169, 23, 10657, 3125, 8361, 19683, 18785, 29, 107442, 31, 65536, 178478, 131361, 94932, 793585, 37, 524649, 1596520, 1439201, 41, 6997770, 43, 4208945, 16302032

Offset: 1

Ilya Gutkovskiy, Mar 30 2025

nonn

Cf. A008472, A055225, A069359, A373458, A373459.

nmax = 45; CoefficientList[Series[Sum[Prime[k] x^Prime[k]/(1 - Prime[k] x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

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

cp's OEIS Frontend

A373458 Expansion of Sum_{p prime} x^p/(1 - p*x^p).

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

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Author

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Crossrefs

Programs

Mathematica

Formula

A382513 Expansion of Sum_{p prime} p * x^p / (1 - p * x^p).

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Author

Keywords

Crossrefs

Programs

Mathematica

Formula