The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A340577 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A340577

Decimal expansion of Product_{primes p == 1 (mod 6)} 1/(1+1/p^2).
(history; published version)

#12 by R. J. Mathar at Sun Aug 21 09:43:32 EDT 2022

STATUS

editing

approved

#11 by R. J. Mathar at Sun Aug 21 09:43:28 EDT 2022

FORMULA

Equals 1.0004615.../A175646, both constants taken from p 27 of arXiv:1008.2537v2. - R. J. Mathar, Aug 21 2022

STATUS

approved

editing

#10 by Vaclav Kotesovec at Fri Jan 15 06:03:28 EST 2021

STATUS

editing

approved

#9 by Vaclav Kotesovec at Fri Jan 15 06:03:12 EST 2021

DATA

9, 6, 7, 5, 5, 0, 4, 0, 2, 5, 1, 9, 5, 6, 1, 8, 8, 6, 6, 0, 9, 4, 7, 0, 7, 7, 0, 4, 3, 9, 0, 6, 7, 7, 3, 0, 0, 1, 5, 2, 4, 9, 1, 2, 9, 6, 0, 3, 0, 4, 3, 8, 6, 3, 5, 6, 3, 0, 2, 3, 9, 8, 0, 8, 4, 0, 6, 8, 7, 3, 9, 5, 1, 6, 3, 8, 3, 9, 9, 9, 4, 6, 1, 6, 0, 5, 4, 1, 7, 8, 7, 3, 7, 7, 4, 2, 2, 3, 6, 8, 7, 5, 9, 8, 21

EXTENSIONS

a(104) corrected by Vaclav Kotesovec, Jan 15 2021

#8 by Vaclav Kotesovec at Fri Jan 15 06:02:37 EST 2021

MATHEMATICA

(* -------------------------------------------------------------------------- *)

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}];

Z[m_, n_, s_] := (w = 1; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = P[m, n, s*w]/w; sumz = sumz + difz; w++]; Exp[sumz]);

$MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Z[6, 1, 4]/Z[6, 1, 2], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021 *)

STATUS

approved

editing

#7 by Peter Luschny at Tue Jan 12 18:23:33 EST 2021

STATUS

proposed

approved

#6 by Jean-François Alcover at Tue Jan 12 08:48:03 EST 2021

STATUS

editing

proposed

#5 by Jean-François Alcover at Tue Jan 12 08:12:44 EST 2021

CROSSREFS

Similar constants: A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340578.

#4 by Jean-François Alcover at Tue Jan 12 07:47:53 EST 2021

CROSSREFS

Similar constants: A175646, A175647, A248930, A248938, A333240, A334826, A335963, A340576, A340578.

#3 by Jean-François Alcover at Tue Jan 12 07:46:53 EST 2021

MATHEMATICA

digits = 105;

precision = digits + 5;

prodeuler[p_, a_, b_, expr_] := Product[If[a <= p <= b, expr, 1], {p, Prime[Range[PrimePi[a], PrimePi[b]]]}];

Lv3[s_] := prodeuler[p, 1, 2^(precision/s), 1/(1 - KroneckerSymbol[-3, p]*p^-s)] // N[#, precision]&;

Lv4[s_] := 2*Im[PolyLog[s, Exp[2*I*Pi/3]]]/Sqrt[3];

Lv[s_] := If[s >= 10000, Lv3[s], Lv4[s]];

gv[s_] := (1 - 3^(-s))*Zeta[s]/Lv[s];

pB = (3/4)*Product[gv[2^n*2]^(2^-(n+1)), {n, 0, 11}] // N[#, precision]&;

pC = (2*Pi^4)/(243*pB*Lv[2]);

RealDigits[pC, 10, digits][[1]](* Most of this code is due to Artur Jasinski *)