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A380497 Euler transform of primorial numbers.

%I A380497 #8 Feb 16 2025 08:34:07
%S A380497 1,2,9,46,314,3072,37641,603510,11148030,249327430,7040987792,
%T A380497 216220333314,7895699690498,321315600822232,13770543972819903,
%U A380497 644232544408157820,33954066516677635554,1994206929690480710244,121461036181617491970561,8111955386813996410196454,574814471423312085719652432
%N A380497 Euler transform of primorial numbers.
%H A380497 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Primorial.html">Primorial</a>.
%F A380497 G.f.: Product_{k>=1} 1 / (1 - x^k)^prime(k)#.
%p A380497 p:= proc(n) option remember; `if`(n<1, 1, p(n-1)*ithprime(n)) end:
%p A380497 a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
%p A380497       add(d*p(d), d=numtheory[divisors](j)), j=1..n)/n)
%p A380497     end:
%p A380497 seq(a(n), n=0..20);  # _Alois P. Heinz_, Jan 25 2025
%t A380497 nmax = 20; CoefficientList[Series[Product[1/(1 - x^k)^Product[Prime[j], {j, k}], {k, 1, nmax}], {x, 0, nmax}], x]
%t A380497 primorial[n_] := Product[Prime[j], {j, 1, n}]; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d primorial[d], {d, Divisors[j]}] a[n - j], {j, 1, n}]/n]; Table[a[n], {n, 0, 20}]
%Y A380497 Cf. A002110, A030009, A030012, A107895, A380498.
%K A380497 nonn
%O A380497 0,2
%A A380497 _Ilya Gutkovskiy_, Jan 25 2025