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A203522 a(n) = v(n+1)/v(n), where v = A203521.

%I A203522 #16 Jan 29 2025 12:24:02
%S A203522 1,5,56,1080,52416,2073600,168537600,9963233280,1122750720000,
%T A203522 234209649623040,22056529195008000,5634088861040640000,
%U A203522 1027073825689514803200,132063784535221862400000,27917533370401003929600000
%N A203522 a(n) = v(n+1)/v(n), where v = A203521.
%H A203522 Robert Israel, <a href="/A203522/b203522.txt">Table of n, a(n) for n = 0..290</a>
%H A203522 Robert Israel, <a href="/A203522/b203522_1.txt">Table of n, a(n) for n = 1..290</a>
%F A203522 a(n) = Product_{i=1..n} (prime(i)+prime(n+1)). - _Robert Israel_, Jan 28 2025
%p A203522 f:= proc(n) local i; mul(ithprime(i)+ithprime(n+1),i=1..n) end proc:
%p A203522 map(f, [$1..30]); # _Robert Israel_, Jan 28 2025
%t A203522 f[j_] := Prime[j]; z = 15;
%t A203522 v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
%t A203522 d[n_] := Product[(i - 1)!, {i, 1, n}] (* A000178 *)
%t A203522 Table[v[n], {n, 1, z}]                (* A203521 *)
%t A203522 Table[v[n + 1]/v[n], {n, 1, z - 1}]   (* A203522 *)
%t A203522 Table[v[n]/d[n], {n, 1, 20}]          (* A203523 *)
%Y A203522 Cf. A203522, A203523, A000040, A093883.
%K A203522 nonn
%O A203522 0,2
%A A203522 _Clark Kimberling_, Jan 03 2012
%E A203522 a(0) = 1 inserted by _Robert Israel_, Jan 29 2025