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%I #21 Dec 14 2022 00:46:40
%S 1,3,5,5,165,165,561,561,12903,170085,170085,170085,170085,170085,
%T 55957965,141368472245,25022219587365,25022219587365,25022219587365,
%U 1776577590702915,1776577590702915,1776577590702915,287890168626762845,4749253940274679,4749253940274679
%N Denominator of asymptotic density of Union{H_p: p is odd prime and p <= n-th prime}, where H_p is {K p(p-1)/2 : K integer}.
%C a(n)= A231808(n)/A231809(n) is the asymptotic density of Union{H_p: p is odd prime and p <= n-th prime}, where H_p:={K p(p-1)/2 : K integer}; a(n) tends to 0.41.. (the asymptotic density of A229307 = Union{H_p: p odd prime}.
%H José María Grau Ribas, <a href="/A231809/b231809.txt">Table of n, a(n) for n = 1..40</a>
%H Jose María Grau, A. M. Oller-Marcen, and J. Sondow, <a href="http://arxiv.org/abs/1309.7941">On the congruence 1^n + 2^n +... + n^n = d (mod n), where d divides n</a>
%e 0, 1/3, 2/5, 2/5, 67/165, 67/165, 230/561, 230/561, 5317/12903, 70307/170085, 70307/170085, 70307/170085, 70307/170085, 70307/170085, 23158993/55957965, 58560723101/141368472245, 10373287618037/25022219587365
%t << DiscreteMath`Combinatorica` (*ver 5.0*)
%t << Combinatorica` (*ver 8.0*)
%t fa[n_] := FactorInteger[n]; lcm[lis_] := lcm[lis] = {aux = 1; Do[aux = LCM[aux, lis[[i]]], {i, 1, Length@lis}]; aux}[[1]]; inclusexclus[lis_] := inclusexclus[lis] =Sum[(-1)^(1 + Length[lis[[i]]])/lcm[lis[[i]]], {i, 1, Length@lis}]; densidad[lis_] := Sum[inclusexclus[KSubsets[lis, i]], {i, 1, Length[lis]}]; lista[n_] := Table[(Prime[i]^2 - Prime[i])/2, {i, 2, n}]; Table[Denominator@densidad[lista[i]], {i, 1, 15}]
%Y Cf. A008837, A229307, A231808.
%K nonn,hard,frac
%O 1,2
%A _José María Grau Ribas_, Nov 13 2013
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