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A325508 Product of primes indexed by the prime exponents of n!.

%I A325508 #8 Nov 20 2019 13:03:11
%S A325508 1,1,2,4,10,20,42,84,204,476,798,1596,3828,7656,12276,24180,36660,
%T A325508 73320,120840,241680,389424,785680,1294440,2588880,3848880,7147920,
%U A325508 11264760,15926040,26057304,52114608,74421648,148843296,187159392,340949280,527531760,926505360
%N A325508 Product of primes indexed by the prime exponents of n!.
%C A325508 The prime indices of a(n) are the signature of n!, which is row n of A115627.
%F A325508 a(n) = A181819(n!).
%F A325508 A001221(a(n)) = A071626(n).
%F A325508 A001222(a(n)) = A000720(n).
%F A325508 A056239(a(n)) = A022559(n).
%F A325508 A003963(a(n)) = A135291(n).
%F A325508 A061395(a(n)) = A011371(n).
%F A325508 A007814(a(n)) = A056171(n).
%F A325508 a(n) = A122111(A307035(n)). - _Antti Karttunen_, Nov 19 2019
%e A325508 We have 7! = 2^4 * 3^2 * 5^1 * 7^1, so a(7) = prime(4)*prime(2)*prime(1)*prime(1) = 84.
%e A325508 The sequence of terms together with their prime indices begins:
%e A325508           1: {}
%e A325508           1: {}
%e A325508           2: {1}
%e A325508           4: {1,1}
%e A325508          10: {1,3}
%e A325508          20: {1,1,3}
%e A325508          42: {1,2,4}
%e A325508          84: {1,1,2,4}
%e A325508         204: {1,1,2,7}
%e A325508         476: {1,1,4,7}
%e A325508         798: {1,2,4,8}
%e A325508        1596: {1,1,2,4,8}
%e A325508        3828: {1,1,2,5,10}
%e A325508        7656: {1,1,1,2,5,10}
%e A325508       12276: {1,1,2,2,5,11}
%e A325508       24180: {1,1,2,3,6,11}
%e A325508       36660: {1,1,2,3,6,15}
%e A325508       73320: {1,1,1,2,3,6,15}
%e A325508      120840: {1,1,1,2,3,8,16}
%e A325508      241680: {1,1,1,1,2,3,8,16}
%t A325508 Table[Times@@Prime/@Last/@If[(n!)==1,{},FactorInteger[n!]],{n,0,30}]
%Y A325508 Cf. A000142, A011371, A022559, A056171, A071626, A076934, A115627, A118914, A122111, A135291, A181819, A307035, A323014, A325272, A325273, A325276, A325509.
%K A325508 nonn
%O A325508 0,3
%A A325508 _Gus Wiseman_, May 08 2019