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A308271 G.f. A(x) satisfies: A(x) = (1/(1 - x)) * A(x^2)*A(x^3)*A(x^5)* ... *A(x^prime(k))* ...

%I A308271 #8 May 17 2019 22:08:02
%S A308271 1,1,2,3,5,7,12,16,24,33,48,64,93,122,169,224,303,395,532,686,907,
%T A308271 1168,1523,1943,2521,3193,4094,5170,6573,8245,10434,13015,16344,20321,
%U A308271 25363,31379,39003,48039,59384,72914,89720,109722,134528,163929,200149,243199,295831,358280
%N A308271 G.f. A(x) satisfies: A(x) = (1/(1 - x)) * A(x^2)*A(x^3)*A(x^5)* ... *A(x^prime(k))* ...
%C A308271 Euler transform of A008480.
%F A308271 G.f.: Product_{k>=1} 1/(1 - x^k)^A008480(k).
%p A308271 g:= proc(n) option remember; (l-> add(i, i=l)!/
%p A308271       mul(i!, i=l))(map(i-> i[2], ifactors(n)[2]))
%p A308271     end:
%p A308271 a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
%p A308271       add(d*g(d), d=numtheory[divisors](j)), j=1..n)/n)
%p A308271     end:
%p A308271 seq(a(n), n=0..60);  # _Alois P. Heinz_, May 17 2019
%t A308271 terms = 47; A[_] = 1; Do[A[x_] = 1/(1 - x) Product[A[x^Prime[k]], {k, 1, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
%Y A308271 Cf. A000040, A008480, A129374, A308272.
%K A308271 nonn
%O A308271 0,3
%A A308271 _Ilya Gutkovskiy_, May 17 2019