|
%I #9 Mar 27 2022 03:14:25
%S 8,0,1,4,6,9,6,9,3,4,2,7,5,7,7,3,3,6,2,2,4,7,0,4,9,3,8,6,8,1,6,9,8,5,
%T 0,7,3,2,7,9,0,5,8,3,3,0,9,3,6,3,2,1,9,6,2,8,9,9,8,2,7,7,6,3,9,4,4,3,
%U 2,9,7,1,9,3,2,0,2,4,0,9,9,6,5,4,8,6,6,9,3,9,0,8,5,4,6,9,3,7,2,2,7,2,8,9,2,2
%N Decimal expansion of Product_{p primes} (1 - 1/p)*(1 + (1 - 1/p^2)*Sum_{k>=1} 1/(p^k + p^(-k-1))).
%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 155 (constant C2).
%H Vaclav Kotesovec, <a href="/A345295/a345295.jpg">Plot of 1/n * Sum_{k=1..n} k*A000010(k)/A057660(k) for n = 1..100000</a>
%F Equals lim_{n->infinity} 1/n * Sum_{k=1..n} k*A000010(k)/A057660(k).
%e 0.80146969342757733622470493868169850732790583309363219628998277639443297...
%t $MaxExtraPrecision = 1000; m = 500; Do[Clear[f]; f[p_] := (1 - 1/p)*(1 + (1 - 1/p^2)*Sum[1/(p^j + p^(-j - 1)), {j, 1, k}]); cc = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x, m + 1]]; Print[f[2]*Exp[N[Sum[Indexed[cc, n]*(PrimeZetaP[n] - 1/2^n), {n, 2, m}], 110]]], {k, 100, 500, 100}]
%Y Cf. A057660, A174405, A345294.
%K nonn,cons
%O 0,1
%A _Vaclav Kotesovec_, Jun 13 2021
|
