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A353065 Euler transform of odd primes.

%I A353065 #8 Apr 21 2022 16:01:11
%S A353065 1,3,11,32,92,239,608,1465,3450,7858,17525,38165,81653,171497,354785,
%T A353065 723084,1454642,2889854,5676607,11031046,21224439,40453596,76428636,
%U A353065 143192339,266172016,491072611,899583306,1636775949,2958900040,5316004485,9494514599
%N A353065 Euler transform of odd primes.
%F A353065 G.f.: Product_{k>=1} 1 / (1 - x^k)^prime(k+1).
%p A353065 a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
%p A353065       d*ithprime(d+1), d=numtheory[divisors](j)), j=1..n)/n)
%p A353065     end:
%p A353065 seq(a(n), n=0..30);  # _Alois P. Heinz_, Apr 21 2022
%t A353065 nmax = 30; CoefficientList[Series[Product[1/(1 - x^k)^Prime[k + 1], {k, 1, nmax}], {x, 0, nmax}], x]
%t A353065 a[0] = 1; a[n_] := a[n] = (1/n) Sum[Sum[d Prime[d + 1], {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 30}]
%Y A353065 Cf. A005380, A030009, A030012, A065091.
%K A353065 nonn
%O A353065 0,2
%A A353065 _Ilya Gutkovskiy_, Apr 21 2022