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A380498 Inverse Euler transform of primorial numbers.

%I A380498 #7 Feb 16 2025 08:34:07
%S A380498 2,3,20,150,1860,24950,444060,8583780,202071920,5992771854,
%T A380498 186947632200,7001535703840,288868991951760,12455290280427090,
%U A380498 587972068547997856,31327583556941402160,1856116108295418943020,113366872636395265380920,7619343577986975410930880,541957669076266398658079700
%N A380498 Inverse Euler transform of primorial numbers.
%H A380498 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Primorial.html">Primorial</a>.
%F A380498 Product_{n>=1} 1 / (1 - x^n)^a(n) = Sum_{n>=0} prime(n)# * x^n.
%p A380498 p:= proc(n) option remember; `if`(n<1, 1, p(n-1)*ithprime(n)) end:
%p A380498 ietr:= proc(p) uses numtheory; (c-> proc(n) option remember;
%p A380498          `if`(n=0, 1, add(mobius(n/d)*c(d), d=divisors(n))/n) end)(
%p A380498           proc(n) option remember; n*p(n)-add(thisproc(j)*p(n-j), j=1..n-1) end)
%p A380498        end:
%p A380498 a:= ietr(p):
%p A380498 seq(a(n), n=1..20);  # _Alois P. Heinz_, Jan 25 2025
%t A380498 primorial[n_] := Product[Prime[j], {j, 1, n}]; b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i] + j - 1, j] b[n - i j, i - 1], {j, 0, n/i}]]]; a[n_] := primorial[n] - b[n, n - 1]; a /@ Range[20]
%Y A380498 Cf. A002110, A030010, A030011, A112354, A380497.
%K A380498 nonn
%O A380498 1,1
%A A380498 _Ilya Gutkovskiy_, Jan 25 2025