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A203530 a(n) = Product_{1 <= i < j <= n} (c(i) + c(j)); c = A002808 = composite numbers.

%I A203530 #16 Jul 23 2017 13:44:53
%S A203530 1,10,1680,5569200,426645273600,1135354270482432000,
%T A203530 129053267560513803386880000,556394398742051964595520667648000000,
%U A203530 99449133623220179596974346585642106880000000000
%N A203530 a(n) = Product_{1 <= i < j <= n} (c(i) + c(j)); c = A002808 = composite numbers.
%C A203530 Each term divides its successor, as in A203530.
%C A203530 It is conjectured that each term is divisible by the corresponding superfactorial, A000178(n); as in A203533.
%C A203530 See A093883 for a guide to related sequences.
%p A203530 c:= proc(n) option remember; local k; if n=1 then 4
%p A203530       else for k from 1+c(n-1) while isprime(k) do od; k fi
%p A203530     end:
%p A203530 a:= n-> mul(mul(c(i)+c(j), i=1..j-1), j=2..n):
%p A203530 seq(a(n), n=1..10);  # _Alois P. Heinz_, Jul 23 2017
%t A203530 t = Table[If[PrimeQ[k], 0, k], {k, 1, 100}];
%t A203530 composite = Rest[Rest[Union[t]]]       (* A002808 *)
%t A203530 f[j_] := composite[[j]]; z = 20;
%t A203530 v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
%t A203530 d[n_] := Product[(i - 1)!, {i, 1, n}]  (* A000178 *)
%t A203530 Table[v[n], {n, 1, z}]                 (* A203530 *)
%t A203530 Table[v[n + 1]/v[n], {n, 1, z - 1}]    (* A203532 *)
%t A203530 Table[v[n]/d[n], {n, 1, 20}]           (* A203533 *)
%Y A203530 Cf. A002808, A203418, A203532, A203533.
%K A203530 nonn
%O A203530 1,2
%A A203530 _Clark Kimberling_, Jan 03 2012
%E A203530 Name edited by _Alois P. Heinz_, Jul 23 2017