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A248930
Decimal expansion of c = 2*Product_{prime p == 3 (mod 4)} (1 - 2/(p*(p-1)^2)), a constant related to the problem of integral Apollonian circle packings.
14
1, 6, 4, 9, 3, 3, 7, 6, 8, 9, 0, 9, 8, 0, 3, 0, 7, 0, 1, 0, 2, 5, 9, 4, 2, 9, 3, 3, 3, 6, 0, 1, 7, 8, 9, 6, 3, 6, 6, 9, 2, 3, 5, 7, 6, 6, 2, 5, 6, 6, 1, 1, 4, 4, 9, 0, 5, 7, 7, 2, 4, 8, 8, 3, 8, 4, 2, 5, 6, 4, 5, 1, 8, 9, 4, 8, 0, 7, 7, 2, 5, 2, 0, 6, 9, 0, 2, 0, 4, 2, 4, 8, 5, 2, 5, 3, 6, 0, 1, 0, 2, 7, 0, 1, 7
OFFSET
1,2
LINKS
Steven R. Finch, Apollonian circles with integer curvatures, p. 6. [Cached copy, with permission of the author]
Elena Fuchs and Katherine Sanden, Some experiments with integral Apollonian circle packings, arXiv:1001.1406 [math.NT] p. 7.
EXAMPLE
1.64933768909803...
MATHEMATICA
kmax = 25; 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}]; c = 2*P[kmax]; RealDigits[c, 10, 15] // 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*Exp[sump], digits]], 10, digits - 1][[1]] (* Vaclav Kotesovec, Jan 16 2021 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
EXTENSIONS
More digits from Vaclav Kotesovec, Jun 27 2020
STATUS
approved