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A297707 a(n) = Product_{k=1..n-1} n!k, where n!k is k-tuple factorial of n.

%I A297707 #22 Dec 02 2018 18:41:21
%S A297707 1,2,18,768,90000,44789760,30494620800,121762322841600,
%T A297707 393644011735296000,5618427494400000000000,
%U A297707 107587910030480590233600000,5951222311476064581656248320000,176804782652901880753915871232000000,69819090744423637487544223697731584000000
%N A297707 a(n) = Product_{k=1..n-1} n!k, where n!k is k-tuple factorial of n.
%C A297707 What is the least n > 2 for which a(n) - prevprime(a(n)) is a composite number? If such a number n exists, it is greater than 250.
%C A297707 The least n for which nextprime(a(n)) - a(n) is a composite number is 158.
%H A297707 Michel Marcus, <a href="/A297707/b297707.txt">Table of n, a(n) for n = 1..100</a>
%F A297707 a(n) = Product_{t=1..n-1} (Product_{k=0..floor((n-1)/t)} (n-t*k)).
%F A297707 a(n) = (n^(n-1))*Product_{k=1..n-1} k^tau(n-k).
%e A297707 a(2) = (2!1) = (2*1) = 2;
%e A297707 a(3) = (3!1)*(3!2) = (3*2*1)*(3*1) = 18;
%e A297707 a(4) = (4!1)*(4!2)*(4!3) = (4*3*2*1)*(4*2)*(4*1) = 768;
%e A297707 a(5) = (5!1)*(5!2)*(5!3)*(5!4) = (5*4*3*2*1)*(5*3*1)*(5*2)*(5*1) = 90000.
%p A297707 b:= proc(n, k) option remember; `if`(n<1, 1, n*b(n-k, k)) end:
%p A297707 a:= n-> mul(b(n, k), k=1..n-1):
%p A297707 seq(a(n), n=1..20);  # _Alois P. Heinz_, Dec 02 2018
%t A297707 Array[(#^(# - 1)) Product[k^DivisorSigma[0, # - k], {k, # - 1}] &, 13] (* _Michael De Vlieger_, Jan 04 2018 *)
%o A297707 (PARI) a(n) = (n^(n-1))*prod(k=1, n-1, k^numdiv(n-k)); \\ _Michel Marcus_, Dec 02 2018
%Y A297707 Cf. A000142, A006882, A007661, A007662, A085157, A085158, A114799, A114800.
%Y A297707 Cf. A114806, A288327, A006990, A033933.
%K A297707 nonn
%O A297707 1,2
%A A297707 _Lechoslaw Ratajczak_, Jan 03 2018