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.

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

Table of n, a(n) for n=1..105.

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

Cf. A002145, A052483, A189226, A189227.

Sequence in context: A179258 A365937 A353769 * A021158 A231535 A019931

Adjacent sequences: A248927 A248928 A248929 * A248931 A248932 A248933

Keyword

nonn,cons

Author

Jean-François Alcover, Oct 17 2014

Extensions

More digits from Vaclav Kotesovec, Jun 27 2020

Status

approved