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

1,2

Links

Table of n, a(n) for n=1..108.

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

Example

1.2569136102101885959492115769468608949404598868075...

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 *)

Prog

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

Crossrefs

Cf. A046073, A105612, A271547.

Sequence in context: A097685 A136369 A305511 * A351830 A244272 A007573

Adjacent sequences: A269469 A269470 A269471 * A269473 A269474 A269475

Keyword

nonn,cons

Author

Jean-François Alcover, Apr 13 2016

Extensions

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

Status

approved