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A325703 If n = prime(i_1)^j_1 * ... * prime(i_k)^j_k, then a(n) is the denominator of the reciprocal factorial sum j_1/i_1! + ... + j_k/i_k!.

%I A325703 #11 Oct 13 2024 11:18:39
%S A325703 1,1,2,1,6,2,24,1,1,6,120,2,720,24,3,1,5040,1,40320,6,24,120,362880,2,
%T A325703 3,720,2,24,3628800,3,39916800,1,120,5040,24,1,479001600,40320,720,6,
%U A325703 6227020800,24,87178291200,120,6,362880,1307674368000,2,12,3,5040,720
%N A325703 If n = prime(i_1)^j_1 * ... * prime(i_k)^j_k, then a(n) is the denominator of the reciprocal factorial sum j_1/i_1! + ... + j_k/i_k!.
%C A325703 Alternatively, if n = prime(i_1) * ... * prime(i_k), then a(n) is the denominator of 1/i_1! + ... + 1/i_k!.
%H A325703 Robert Israel, <a href="/A325703/b325703.txt">Table of n, a(n) for n = 1..3168</a>
%H A325703 Gus Wiseman, <a href="/A051908/a051908.txt">Sequences counting and ranking integer partitions by their reciprocal sums</a>
%F A325703 a(n) = A318574(A325709(n)).
%p A325703 f:= proc(n) local F,t;
%p A325703     F:= ifactors(n)[2];
%p A325703     denom(add(t[2]/numtheory:-pi(t[1])!,t=F))
%p A325703 end proc:
%p A325703 map(f, [$1..100]); # _Robert Israel_, Oct 13 2024
%t A325703 Table[Total[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>k/PrimePi[p]!]],{n,100}]//Denominator
%Y A325703 Factorial numbers: A000142, A002982, A011371, A022559, A071626, A076934, A108731, A325272, A325508, A325709.
%Y A325703 Reciprocal sum: A002966, A316855, A316856, A316857, A318573, A318574, A325618, A325619, A325620, A325621, A325622, A325623, A325624, A325704.
%K A325703 nonn,frac,look
%O A325703 1,3
%A A325703 _Gus Wiseman_, May 18 2019