A352005 -id:A352005 - cp's OEIS frontend

A352014 a(n) = Sum_{p|n, p prime} (-1)^(n/p+1) * (n-1)!/(p-1)!.

0, 1, 1, -6, 1, 60, 1, -5040, 20160, 347760, 1, -59875200, 1, 6218372160, 47221574400, -1307674368000, 1, 177843714048000, 1, -126713646259200000, 1219830034655232000, 51090928092415411200, 1, -38778025108327464960000, 25852016738884976640000

Offset: 1

Seiichi Manyama, Feb 28 2022

sign

Cf. A352005, A352012, A352013.

a[n_] := Sum[(-1)^(n/p + 1)*(n - 1)!/(p - 1)!, {p, FactorInteger[n][[;; , 1]]}]; a[1] = 0; Array[a, 25] (* Amiram Eldar, Oct 04 2023 *)

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

my(N=40, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=1, N, isprime(k)*log(1+x^k)/k!))))

E.g.f.: Sum_{p prime} log(1+x^p)/p!.

A352003 Expansion of e.g.f. Product_{k>=1} (1 + x^prime(k))^(1/prime(k)).

Original entry on oeis.org

1, 0, 1, 2, -3, 44, -35, 1014, -1127, 46808, 153081, 3240170, -30922859, 443621892, 331421077, 121899383774, 691635821745, 19657393214384, 424491327098353, 2132527815161298, -2864544697983059, 3885322666246386140, 22621061924336157261, 882556261002776755142

Offset: 0

Seiichi Manyama, Feb 28 2022

sign

Cf. A168243, A305614, A318913, A352005.

my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+isprime(k)*x^k)^(1/k))))

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, sumdiv(k, d, isprime(d)*(-1)^(k/d+1))*x^k/k))))

E.g.f.: exp( Sum_{k>=1} A305614(k)*x^k/k ) where A305614(k) = Sum_{p|k, p prime} (-1)^(k/p+1).

cp's OEIS Frontend

A352014 a(n) = Sum_{p|n, p prime} (-1)^(n/p+1) * (n-1)!/(p-1)!.

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A352003 Expansion of e.g.f. Product_{k>=1} (1 + x^prime(k))^(1/prime(k)).

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Author

Keywords

Crossrefs

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PARI

PARI

Formula