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A231808 Numerator 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}.

%I A231808 #37 Feb 25 2025 01:46:08
%S A231808 0,1,2,2,67,67,230,230,5317,70307,70307,70307,70307,70307,23158993,
%T A231808 58560723101,10373287618037,10373287618037,10373287618037,
%U A231808 736719736564627,736719736564627,736719736564627,119433196256360189,1970856524120023,1970856524120023,1970856524120023
%N A231808 Numerator 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 A231808 a(n)/A231809(n) is the 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}; a(n) tends to 0.41.. (the asymptotic density of A229307 = Union{H_p: p odd prime}).
%H A231808 José María Grau Ribas, <a href="/A231808/b231808.txt">Table of n, a(n) for n = 1..40</a>
%H A231808 Jose María Grau, A. M. Oller-Marcen, and Jonathan 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>, arXiv:1309.7941 [math.NT], 2013-2014.
%e A231808 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 A231808 << DiscreteMath`Combinatorica` (*ver 5.0*)
%t A231808 << Combinatorica` (*ver 8.0*)
%t A231808 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[Numerator@densidad[lista[i]], {i, 1, 15}]
%Y A231808 Cf. A008837, A229307, A231809.
%K A231808 nonn,hard,frac
%O A231808 1,3
%A A231808 _José María Grau Ribas_, Nov 13 2013