A248938 Decimal expansion of beta = G^2*(2/3)*Product_{prime p == 3 (mod 4)} (1 - 2/(p*(p-1)^2)) (where G is Catalan's constant), a constant related to the problem of integral Apollonian circle packings.
4, 6, 1, 2, 6, 0, 9, 0, 8, 6, 1, 3, 8, 6, 1, 3, 0, 3, 3, 2, 8, 5, 2, 9, 8, 4, 6, 4, 2, 4, 6, 0, 7, 5, 1, 5, 8, 0, 1, 3, 8, 3, 4, 4, 3, 7, 6, 5, 8, 8, 2, 0, 6, 3, 0, 0, 7, 0, 3, 9, 7, 7, 5, 1, 9, 0, 7, 1, 2, 8, 1, 6, 0, 7, 2, 2, 0, 7, 4, 9, 8, 3, 7, 9, 1, 0, 4, 2, 6, 0, 7, 2, 6, 2, 1, 4, 8, 0, 7, 2, 3, 1, 6, 3, 1, 6
Offset: 0
Examples
0.4612609086138613...
Links
- Steven R. Finch, Apollonian circles with integer curvatures, p. 6. [Cached copy, with permission of the author]
- Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 256.
- Elena Fuchs and Katherine Sanden, Some experiments with integral Apollonian circle packings, arXiv:1001.1406 [math.NT] p. 7.
- Peter Sarnak, Integral Apollonian Packings, Princeton MAA Lecture - January, 2009, p. 21.
Programs
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Mathematica
kmax = 25; Clear[P]; Do[P[k] = Product[p = Prime[n]; If[Mod[p, 4] == 3 , 1 - 2/(p*(p - 1)^2) // N[#, 40]&, 1], {n, 1, 2^k}]; Print["P(", k, ") = ", P[k]], {k, 10, kmax}]; beta = Catalan^2*(2/3)*P[kmax]; RealDigits[beta, 10, 16] // First (* -------------------------------------------------------------------------- *) $MaxExtraPrecision = 1000; digits = 121; f[p_] := (1 - 2/(p*(p - 1)^2)); coefs = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, 1000}], x]]; S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums); P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]] * S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}]; m = 2; sump = 0; difp = 1; While[Abs[difp] > 10^(-digits - 5) || difp == 0, difp = coefs[[m]]*P[4, 3, m]; sump = sump + difp; m++]; RealDigits[Chop[N[2*Catalan^2/3 * Exp[sump], digits]], 10, digits - 1][[1]] (* Vaclav Kotesovec, Jan 16 2021 *)
Extensions
More digits from Vaclav Kotesovec, Jun 27 2020