A030009 Euler transform of primes.

1, 2, 6, 15, 37, 85, 192, 414, 879, 1816, 3694, 7362, 14480, 28037, 53644, 101379, 189587, 350874, 643431, 1169388, 2108045, 3770430, 6694894, 11804968, 20679720, 35999794, 62298755, 107198541, 183462856, 312357002, 529173060, 892216829, 1497454396, 2502190992

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A007441.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*ithprime(d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  #  Alois P. Heinz, Sep 06 2008

Mathematica

a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*Prime[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 16 2014, after Alois P. Heinz *)

PARI

a(n)=if(n<0,0,polcoeff(prod(i=1,n,(1-x^i)^-prime(i),1+x*O(x^n)),n))

SageMath

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: nth_prime(n))
print([b(n) for n in range(37)]) # Peter Luschny, Nov 11 2020

Formula

G.f.: Product_{n>=1} (1-x^n)^(-prime(n)).