A319113 -id:A319113 - cp's OEIS frontend

A334370 Expansion of e.g.f. Product_{k>=1} (1 + x^prime(k) / prime(k)!).

1, 0, 1, 1, 0, 11, 0, 22, 56, 36, 2640, 1, 8712, 79, 72436, 360465, 48608, 49008961, 794376, 4232764, 7753140, 942565890, 18198334, 14799637777, 10577976, 366619314900, 2785137222400, 1475339135400, 1065920156634060, 3765722000041, 5869315258699050

Offset: 0

Ilya Gutkovskiy, May 11 2020

nonn

a(n) is the number of functions f:[n]-> [n] such that the number of elements that are mapped to i is either 0 or the i-th prime. a(5) = 11: (33333), (11222), (12122), (12212), (12221), (21122), (21212), (21221), (22112), (22121), (22211). - Alois P. Heinz, Jul 18 2023

Alois P. Heinz, Table of n, a(n) for n = 0..836

Cf. A000040, A000720, A007837, A032310, A115278, A178682, A190476, A319113, A364344.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+
      (p-> `if`(p>n, 0, b(n-p, i-1)*binomial(n, p)))(ithprime(i))))
    end:
a:= n-> b(n, numtheory[pi](n)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 18 2023

nmax = 30; CoefficientList[Series[Product[(1 + x^Prime[k]/Prime[k]!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[DivisorSum[k, -#/(-#!)^(k/#) &, PrimeQ[#] &] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 30}]

my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, 1+isprime(k)*x^k/k!))) \\ Seiichi Manyama, Feb 27 2022

A319112 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^prime(k)/prime(k)).

Original entry on oeis.org

1, 0, 1, 2, 6, 44, 170, 1644, 7448, 72624, 653112, 8510160, 62704752, 1324662624, 10772812752, 167386388064, 2413326453120, 52610523489024, 597065112874368, 18066985168806144, 212119023906342144, 4734822914239173120, 100734270778298352384, 2818116390408742291968, 48201015565806837709824

Offset: 0

Ilya Gutkovskiy, Sep 10 2018

nonn

Cf. A002098, A007841, A319113.

seq(n!*coeff(series(mul(1/(1-x^ithprime(k)/ithprime(k)),k=1..100),x=0,25),x,n),n=0..24); # Paolo P. Lava, Jan 09 2019

nmax = 24; CoefficientList[Series[Product[1/(1 - x^Prime[k]/Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 24; CoefficientList[Series[Exp[Sum[Sum[Boole[PrimeQ[d]] d^(1 - k/d), {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[Sum[Boole[PrimeQ[d]] d^(1 - k/d), {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 24}]

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, 1-isprime(k)*x^k/k))) \\ Seiichi Manyama, Feb 27 2022

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

cp's OEIS Frontend

A334370 Expansion of e.g.f. Product_{k>=1} (1 + x^prime(k) / prime(k)!).

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