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Euler transform of primes.
10

%I #22 Nov 11 2020 09:08:47

%S 1,2,6,15,37,85,192,414,879,1816,3694,7362,14480,28037,53644,101379,

%T 189587,350874,643431,1169388,2108045,3770430,6694894,11804968,

%U 20679720,35999794,62298755,107198541,183462856,312357002,529173060,892216829,1497454396,2502190992

%N Euler transform of primes.

%H Alois P. Heinz, <a href="/A030009/b030009.txt">Table of n, a(n) for n = 0..1000</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

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

%p with(numtheory):

%p a:= proc(n) option remember; `if`(n=0, 1, add(add(

%p d*ithprime(d), d=divisors(j))*a(n-j), j=1..n)/n)

%p end:

%p seq(a(n), n=0..40); # _Alois P. Heinz_, Sep 06 2008

%t 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_ *)

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

%o (SageMath) # uses[EulerTransform from A166861]

%o b = EulerTransform(lambda n: nth_prime(n))

%o print([b(n) for n in range(37)]) # _Peter Luschny_, Nov 11 2020

%Y Cf. A007441.

%K nonn

%O 0,2

%A _N. J. A. Sloane_.