A269472 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.

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, 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, 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

Offset: 1

Author

Jean-François Alcover, Apr 13 2016

Keywords

• nonn
• cons

Examples

			1.2569136102101885959492115769468608949404598868075...
		

Links

• Steven R. Finch and Pascal Sebah, Squares and Cubes Modulo n, arXiv:math/0604465 [math.NT], 2006-2016.

Crossrefs

Cf. A046073, A105612, A271547.

Programs

Mathematica

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]]
(* Second program: *)
$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 *)

PARI

sqrt(prodeulerrat((1-(p^2+2)/(2*(p^2+1)*(p+1)))^2/(1-1/p))) \\ Amiram Eldar, May 29 2021

Extensions

Formula in name and last digit corrected by Vaclav Kotesovec, Jun 19 2020