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}.

0, 1, 2, 2, 67, 67, 230, 230, 5317, 70307, 70307, 70307, 70307, 70307, 23158993, 58560723101, 10373287618037, 10373287618037, 10373287618037, 736719736564627, 736719736564627, 736719736564627, 119433196256360189, 1970856524120023, 1970856524120023, 1970856524120023

Offset

1,3

Comments

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 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}.

Links

José María Grau Ribas, Table of n, a(n) for n = 1..40

Jose María Grau, A. M. Oller-Marcen, and J. Sondow, On the congruence 1^n + 2^n +... + n^n = d (mod n), where d divides n

Example

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, ....

Mathematica

<< DiscreteMath`Combinatorica` (*ver 5.0*)
<< Combinatorica` (*ver 8.0*)
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}]

Crossrefs

Cf. A008837, A229307, A231809.

Sequence in context: A286377 A187024 A274477 * A306063 A028372 A130678

Adjacent sequences: A231805 A231806 A231807 * A231809 A231810 A231811

Keyword

nonn,hard,frac

Author

José María Grau Ribas, Nov 13 2013

Status

approved