A301430 -id:A301430 - cp's OEIS frontend

A301429 Decimal expansion of an analog of the Landau-Ramanujan constant for Loeschian numbers.

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

Offset: 0

Michel Waldschmidt, Mar 21 2018

This is the decimal expansion of the number alpha such that the number of positive integers <= N which are represented by the quadratic form x^2 + xy + y^2 is asymptotic to alpha*N/sqrt(log(N)).

			0.638909405445343882254942674928245093754975508...

S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, p. 99 (K3).

Peter Luschny, Table of n, a(n) for n = 0..1000 (terms 0..105 from Vaclav Kotesovec).
Salma Ettahri, Olivier Ramaré, and Léon Surel, Fast multi-precision computation of some Euler products, arXiv:1908.06808 [math.NT], 2019.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 204.
Étienne Fouvry, Claude Levesque, and Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017 and Acta Arithmetica, online 15 March 2018.
Olivier Ramaré, S. Ettahri, and L. Surel, Fast multi-precision computation of some Euler products, Mathematics of Computation (2021) hal-03381427.
StackExchange, Iterative calculation of a number-theoretical constant, Mar 24 2018.

Cf. A003136, A003627, A064533, A301430, A333240.

Digits:= 1000: A:= 2^(-1/2)*3^(-1/4):
for t to 40000 do p:= ithprime(t): if `mod`(p, 3) = 2 then
A:= evalf(A/(1-1/p^2)^(1/2)) end if end do: A;
# Alternative:
z := n -> Zeta(n)/Im(polylog(n, (-1)^(2/3))):
x := n -> (z(2^n)*(3^(2^n)-1)*sqrt(3)/2)^(1/2^n)/3:
evalf(sqrt(mul(x(n), n=1..8))/12^(1/4), 110); # Peter Luschny, Jan 17 2021

digits = 106;
precision = digits + 10;
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];
pgv = Product[gv[2^n*2]^(2^-(n + 1)), {n, 0, 11}] // N[#, precision]&;
RealDigits[Sqrt[pgv]/12^(1/4), 10, digits][[1]]
(* Jean-François Alcover, Jan 12 2021, after PARI code due to Artur Jasinski *)
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[Pi * Sqrt[2] / (3^(7/4) * Sqrt[Z[3, 1, 2]]), digits]], 10, digits-1][[1]]
(* Vaclav Kotesovec, Jan 15 2021 *)
z[n_] := Zeta[n]/Im[PolyLog[n, (-1)^(2/3)]];
x[n_] := (z[2^n] (3^(2^n) - 1) Sqrt[3]/2)^(1/2^n)/3;
N[Sqrt[Product[x[n], { n, 8}]]/12^(1/4), 110] (* Peter Luschny, Jan 17 2021 *)

Equals 2^(-1/2)*3^(-1/4)*Product_{p == 2 (mod 3), p prime} (1 - p^(-2))^(-1/2).
One can base the definition on p(n) = A003627(n). Setting r(n) = (Product_{k=1..n} p(k)^2) / (Product_{k=1..n} (p(k)^2 - 1)) the rational sequence r(n) starts 4/3, 25/18, 605/432, 174845/124416, ... -> L. Then A301429 = sqrt(L)/12^(1/4). - Peter Luschny, Mar 29 2018 [This L is now A333240. - Peter Luschny, Jan 14 2021]
Equals Pi*sqrt(2) / (3^(7/4) * sqrt(A175646)). - Vaclav Kotesovec, May 12 2020
Equals 12^(-1/4)*Product_{n>=0} a(-n-2)*b(2^(n+1))^(2^(-n-2)) where a(n) = 3^(2^(n-1))*(1/2-3^(-2^(-n-1))/2)^(2^n) and b(n) = zeta(n)/Im(polylog(n, (-1)^(2/3))). - Peter Luschny, Jan 14 2021

Offset corrected by Vaclav Kotesovec, Mar 25 2018
a(6)-a(10) from Peter Luschny, Mar 29 2018
More digits from Ettahri article added by Vaclav Kotesovec, May 12 2020
More digits from Vaclav Kotesovec, Jun 27 2020

A340127 Decimal expansion of Product_{primes p == 4 (mod 5)} p^2/(p^2-1).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jan 15 2021

			1.0049603239222975589937496248102521847955102941880228801995283785215071277...

Vaclav Kotesovec, Table of n, a(n) for n = 1..501
S. Ettahri, O. Ramare, L. Surel, Fast multi-precision computation of some Euler products, arxiv:1908.06808 (2019), Section 9.
Steven Finch and Pascal Sebah, Residue of a Mod 5 Euler Product, arXiv:0912.3677 [math.NT], 2009 (C(5,n) = mu(n,5) formulas p.2).
Alessandro Languasco and Alessandro Zaccagnini, Computation of the Mertens constants mod q; 3 <= q <= 100, (2007) (GP-PARI procedure 100 digits accuracy).
Alessandro Languasco and Alessandro Zaccagnini, On the constant in the Mertens product for arithmetic progressions. I. Identities, Funct. Approx. Comment. Math. Volume 42, Number 1 (2010), 17-27.
For other links see A340711.

Cf. A004618, A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340577, A340578, A340628, A340628, A340004.

(* Using Vaclav Kotesovec's function Z from A301430. *)
$MaxExtraPrecision = 1000; digits = 121;
digitize[c_] := RealDigits[Chop[N[c, digits]], 10, digits - 1][[1]];
digitize[Z[5, 4, 2]]

Equals (1/C(5,4))*Pi*sqrt(3*C(5,1)*C(5,2)*C(5,3)/(5*C(5,4)*log(2+sqrt(5)))).
for definitions of Mertens constants C(5,n) see A. Languasco and A. Zaccagnini 2010.
for high precision numerical values C(5,n) see A. Languasco and A. Zaccagnini 2007.
C(5,1)=1.225238438539084580057609774749220527540595509391649938767...
C(5,2)=0.546975845411263480238301287430814037751996324100819295153...
C(5,3)=0.8059510404482678640573768602784309320812881149390108979348...
C(5,4)=1.29936454791497798816084001496426590950257497040832966201678...
Equals (1/C(5,4)^2)*Pi*sqrt(3*exp(-gamma)/(4*log(2 + sqrt(5)))), where gamma is the Euler-Mascheroni constant A001620.
Equals Sum_{k>=1} 1/A004618(k)^2. - Amiram Eldar, Jan 24 2021

A340711 Decimal expansion of Product_{primes p == 3 (mod 5)} (p^2+1)/(p^2-1).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jan 16 2021

			1.273986613206833925158...

Vaclav Kotesovec, Table of n, a(n) for n = 1..500
Salma Ettahri, Olivier Ramaré, and Léon Surel, Fast multi-precision computation of some Euler product, arXiv:1908.06808 [math.NT], 2019, p. 20.
Steven Finch, Quartic and Octic Characters Modulo n, arXiv:1008.2547 [math.NT], 2007-2010 p. 11 (formula on kappa(5) and kappa(-5)).
Steven Finch, Greg Martin and Pascal Sebah, Roots of unity and nullity modulo n, Proc. Amer. Math. Soc. Volume 138, Number 8, August 2010, pp. 2729-2743.
Steven Finch and Pascal Sebah, Residue of a Mod 5 Euler Product, arXiv:0912.3677 [math.NT], 2009 (formulas).
Alessandro Languasco and Alessandro Zaccagnini, A note on Mertens' formula for arithmetic progressions, Journal of Number Theory Volume 127, Issue 1, (2007), 37-46.
Alessandro Languasco and Alessandro Zaccagnini, On the constant in the Mertens product for arithmetic progressions. II: Numerical values, Math. Comp. 78 (2009), 315-326.
Alessandro Languasco and Alessandro Zaccagnini, On the constant in the Mertens product for arithmetic progressions. I. Identities, Funct. Approx. Comment. Math. Volume 42, Number 1 (2010), 17-27.
Alessandro Languasco and Alessandro Zaccagnini, Computation of the Mertens constants - more than 100 correct digits, (2007), 1-134 (digital data).
Alessandro Languasco and Alessandro Zaccagnini, Computation of the Mertens constants mod q; 3 <= q <= 100, (2007) (GP-PARI procedure 100 digits accuracy).
Richard J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.
G. Niklasch, Some number-theoretical constants arising as products of rational functions of p over primes
Doug S. Phillips and Peter Zvengrowski, Convergence of Dirichlet Series and Euler Products, Contributions Section of Natural Mathematical and Biotechnical Sciences 38(2):153 (2017).

Cf. A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340577, A340578, A340628, A340629, A340004, A340127, A340710.

(* Using Vaclav Kotesovec's function Z from A301430. *)
$MaxExtraPrecision = 1000; digits = 121;
digitize[c_] := RealDigits[Chop[N[c, digits]], 10, digits - 1][[1]];
digitize[1/(Z[5, 3, 4]/Z[5, 3, 2]^2)]

D = Product_{primes p == 0 (mod 5)} (p^2+1)/(p^2-1) = 13/12.
E = Product_{primes p == 1 (mod 5)} (p^2+1)/(p^2-1) = A340629.
F = Product_{primes p == 2 (mod 5)} (p^2+1)/(p^2-1) = A340710.
G = Product_{primes p == 3 (mod 5)} (p^2+1)/(p^2-1) = this constant.
H = Product_{primes p == 4 (mod 5)} (p^2+1)/(p^2-1) = A340628.
D*E*F*G*H = 5/2.
E*F*G*H = 30/13.
D*E*H = sqrt(5)/2.
D*F*G = 13*sqrt(5)/12.
F*G = sqrt(5).
E*H = 6*sqrt(5)/13.
Equals Sum_{q in A004617} 2^A001221(q)/q^2. - R. J. Mathar, Jan 27 2021

A340665 Decimal expansion of Product_{primes p == 3 (mod 5)} p^2/(p^2-1).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jan 15 2021

			1.135764878668921626868643009472082289511936413...

Vaclav Kotesovec, Table of n, a(n) for n = 1..501
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions..., arXiv:1008.2547 Zeta_{m=5,n=3}(s=2).
For links see A340628.

Cf. A004617, A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340577, A340578, A340628, A340628, A340004, A340127.

(* Using Vaclav Kotesovec's function Z from A301430. *)
$MaxExtraPrecision = 100; digits = 50; (* Adjust as needed. *)
digitize[c_] := RealDigits[Chop[N[c, digits+10]], 10, digits][[1]];
digitize[Z[5, 3, 2]]

Equals Sum_{k>=1} 1/A004617(k)^2. - Amiram Eldar, Jan 24 2021

A340794 Decimal expansion of Product_{primes p == 2 (mod 5)} p^2/(p^2-1).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jan 21 2021

			1.36857205387664908586076389048310999017020782888589520500850402955633118881...

Vaclav Kotesovec, Table of n, a(n) for n = 1..501
For links see A340711.

Cf. A004616, A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340127, A340576, A340577, A340578, A340628, A340629, A340665, A340710, A340711.

(* Using Vaclav Kotesovec's function Z from A301430. *)
$MaxExtraPrecision = 1000; digits = 121;
digitize[c_] := RealDigits[Chop[N[c, digits]], 10, digits - 1][[1]];
digitize[Z[5, 2, 2]]

I = Product_{primes p == 0 (mod 5)} p^2/(p^2-1) = 25/24.
J = Product_{primes p == 1 (mod 5)} p^2/(p^2-1) = A340004.
K = Product_{primes p == 2 (mod 5)} p^2/(p^2-1) = this constant.
L = Product_{primes p == 3 (mod 5)} p^2/(p^2-1) = A340665.
M = Product_{primes p == 4 (mod 5)} p^2/(p^2-1) = A340127.
I*J*K*L*M = Pi^2/6 = zeta(2).
J*K*L*M = 4*Pi^2/25.
M = (Pi/2)*C(5,4)^(-2)*exp(-gamma/2)*sqrt(3/log(2+sqrt(5))), where gamma is the Euler-Mascheroni constant A001620 and C(5,4) is the Mertens constant = 1.29936454791497798816084...
Equals Sum_{k>=1} 1/A004616(k)^2. - Amiram Eldar, Jan 24 2021

A340710 Decimal expansion of Product_{primes p == 2 (mod 5)} (p^2+1)/(p^2-1).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jan 16 2021

			1.7551738411687377766074721228405237...

Vaclav Kotesovec, Table of n, a(n) for n = 1..500
For links see A340711.

Cf. A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340577, A340578, A340628, A340629, A340004, A340127, A340711.

(* Using Vaclav Kotesovec's function Z from A301430. *)
$MaxExtraPrecision = 1000; digits = 121;
digitize[c_] := RealDigits[Chop[N[c, digits]], 10, digits - 1][[1]];
digitize[1/(Z[5, 2, 4]/Z[5, 2, 2]^2)]

D = Product_{primes p == 0 (mod 5)} (p^2+1)/(p^2-1) = 13/12.
E = Product_{primes p == 1 (mod 5)} (p^2+1)/(p^2-1) = A340629.
F = Product_{primes p == 2 (mod 5)} (p^2+1)/(p^2-1) = this constant.
G = Product_{primes p == 3 (mod 5)} (p^2+1)/(p^2-1) = A340711.
H = Product_{primes p == 4 (mod 5)} (p^2+1)/(p^2-1) = A340628.
D*E*F*G*H = 5/2.
E*F*G*H = 30/13.
D*E*H = sqrt(5)/2.
D*F*G = 13*sqrt(5)/12.
F*G = sqrt(5).
E*H = 6*sqrt(5)/13.
Formulas by Pascal Sebah, Jan 20 2021: (Start)
Let g = sqrt(Cl2(2*Pi/5)^2+Cl2(4*Pi/5)^2) = 1.0841621352693895..., where Cl2 is the Clausen function of order 2.
E = 15*sqrt(65)*g/(13*Pi^2).
H = 6*sqrt(13)*Pi^2/(195*g). (End)
Equals Sum_{q in A004616} 2^A001221(q)/q^2. - R. J. Mathar, Jan 27 2021

A340866 Decimal expansion of the Mertens constant C(5,4).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jan 24 2021

Data taken from Alessandro Languasco and Alessandro Zaccagnini 2007 p. 4.

			1.299364547914977988160840014964265909502574970408329662016...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.2 Meissel-Mertens constants (pp. 94-95).

Vaclav Kotesovec, Table of n, a(n) for n = 1..497
Alessandro Languasco and Alessandro Zaccagnini, On the constant in the Mertens product for arithmetic progressions. II: Numerical values, Math. Comp. 78 (2009), 315-326.
Alessandro Languasco and Alessandro Zaccagnini, Computation of the Mertens constants - more than 100 correct digits, (2007). [in this table on page 4, the last correct digit is a(108), beyond the level there certified. - _Vaclav Kotesovec_, Jan 26 2021]
Alessandro Languasco and Alessandro Zaccagnini, On the constant in the Mertens product for arithmetic progressions. I. Identities, Funct. Approx. Comment. Math. Volume 42, Number 1 (2010), 17-27.
For other links see A340711.

Cf. A340004, A340127, A340839, A340711.

(* Using Vaclav Kotesovec's function Z from A301430. *)
$MaxExtraPrecision = 1000; digits = 121;
digitize[c_] := RealDigits[Chop[N[c, digits]], 10, digits - 1][[1]];
digitize[(13*Pi^2 / (24*Sqrt[5] * Exp[EulerGamma] * Log[(1 + Sqrt[5])/2]) * Z[5, 1, 2]^2 / (Z[5, 1, 4] * Z[5, 4, 4]))^(1/4)]

Equals A340839*5^(1/4)*sqrt(A340004/(2*A340127)).

Equals (13*Pi^2/(24*sqrt(5)*exp(gamma)*log((1+sqrt(5))/2))*A340629/A340809)^(1/4). - Vaclav Kotesovec, Jan 25 2021

Corrected by Vaclav Kotesovec, Jan 25 2021

More digits from Vaclav Kotesovec, Jan 26 2021

A333239 Decimal expansion of lim_{n->infinity} (Product_{k=1..n} p(k)^2 / Product_{k=1..n} (p(k)^2 - 1)) where p(k) = A167135(k) are the primes with p mod 12 != 1.

Original entry on oeis.org

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

Offset: 1

Peter Luschny, May 13 2020

			1.63250540290251307901036625893553760497201130044326292653142032442675746272540...

Cf. A167135, A301430, A333304, A333240.

A301430 = sqrt(A333239)*A333304.

A340552 Decimal expansion of Product_{primes p == 5, 7, 11 (mod 12)} 1/(1 - 1/p^2).

Original entry on oeis.org

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

Offset: 0

Peter Luschny, Jan 19 2021

			1.0883369352683420526735775059570250699813408669621752843542802162845...

Salma Ettahri, Olivier Ramaré, and Léon Surel, Fast multi-precision computation of some Euler products, arXiv:1908.06808 [math.NT], 2019.
Étienne Fouvry, Claude Levesque, and Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017 and Acta Arithmetica, online 15 March 2018.

A301430.

(* Using Vaclav Kotesovec's function Z from A301430. *)
$MaxExtraPrecision = 1000; digits = 90;
digitize[c_] := RealDigits[Chop[N[c, digits]], 10, digits - 1][[1]];
digitize[Z[12, 5, 2] Z[12, 7, 2] Z[12, 11, 2]]

A340552^(1/2) = A301430 / (3^(1/4)*Pi^(1/2)*log(2+sqrt(3))^(1/4)/(2^(5/4)* Gamma(1/4))), see É. Fouvry et al.

A333304 Decimal expansion of (arccosh(2)/6)^(1/4)Gamma(3/4)/(2sqrt(Pi)).

Original entry on oeis.org

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

Offset: 0

Peter Luschny, May 13 2020

			0.236610505606501753989409239546869068431564857907961545850713735434682468877...

Cf. A301430, A333239.

(log(2 + sqrt(3))/6)^(1/4)*GAMMA(3/4)/(2*sqrt(Pi)): evalf(%, 99);

c := (Gamma[3/4] (Log[2 + Sqrt[3]] / 6)^(1/4))/(2 Sqrt[Pi]); N[c, 100]

Equals A301430 / sqrt(A333239).

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A333304 Decimal expansion of (arccosh(2)/6)^(1/4)Gamma(3/4)/(2sqrt(Pi)).