|
%I #33 May 29 2021 04:10:19
%S 1,2,5,6,9,1,3,6,1,0,2,1,0,1,8,8,5,9,5,9,4,9,2,1,1,5,7,6,9,4,6,8,6,0,
%T 8,9,4,9,4,0,4,5,9,8,8,6,8,0,7,5,0,8,7,6,7,7,9,8,5,7,1,8,1,9,3,4,7,5,
%U 1,8,2,3,8,4,5,7,4,5,4,1,4,8,7,5,5,3,9,7,5,4,8,9,7,8,6,4,9,1,1,5,7,6,4,5,0,9,9,6
%N Decimal expansion of Product_{p prime} (1-(p^2+2)/(2(p^2+1)(p+1))) / sqrt(1-1/p), a constant related to the asymptotic average number of squares modulo n.
%H Steven R. Finch and Pascal Sebah, <a href="https://arxiv.org/abs/math/0604465">Squares and Cubes Modulo n</a>, arXiv:math/0604465 [math.NT], 2006-2016.
%e 1.2569136102101885959492115769468608949404598868075...
%t digits = 104; m0 = 100; Clear[s]; s[m_] := s[m] = Sum[(1 + 2*(-1)^n - 4*(-1)^n*ChebyshevT[n, 1/4] + 4*Switch[Mod[n, 4], 2, -1, 3, 0, 0, 1, 1, 0])/(2*n) PrimeZetaP[n], {n, 2, m}] // N[#, digits]& // Exp; s[m0]; s[m = 2 m0]; While[RealDigits[s[m], 10, digits] != RealDigits[s[m/2], 10, digits], m = 2 m; Print[m]]; RealDigits[s[m]][[1]]
%t (* Second program: *)
%t $MaxExtraPrecision = 1000; Clear[f]; f[p_] := (1 - (p^2 + 2)/(2 (p^2 + 1) (p + 1)))/ Sqrt[1 - 1/p]; Do[c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; Print[f[2] * Exp[N[Sum[Indexed[c, n]*(PrimeZetaP[n] - 1/2^n), {n, 2, m}], 112]]], {m, 100, 1000, 100}] (* _Vaclav Kotesovec_, Jun 19 2020 *)
%o (PARI) sqrt(prodeulerrat((1-(p^2+2)/(2*(p^2+1)*(p+1)))^2/(1-1/p))) \\ _Amiram Eldar_, May 29 2021
%Y Cf. A046073, A105612, A271547.
%K nonn,cons
%O 1,2
%A _Jean-François Alcover_, Apr 13 2016
%E Formula in name and last digit corrected by _Vaclav Kotesovec_, Jun 19 2020
|
