A167135 -id:A167135 - cp's OEIS frontend

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.

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.

A301430 Decimal expansion of an analog of the Landau-Ramanujan constant for Loeschian numbers which are sums of two squares.

Original entry on oeis.org

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

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 sums of two squares and are also represented by the quadratic form x^2 + xy + y^2 is asymptotic to alpha*N*(log(N))^(-3/4).

Based on the constants Zeta(m=12,n=5,s=2) = 1.0482019036007..., Zeta(m=12,n=7,s=2) = 1.0262021468... and Zeta(m=12,n=11,s=2) = 1.01177863 ... read from arXiv:1008.2547 we have Product_{p == 5, 7, 11(mod 12)} (1-1/p^2)^(-1/2) = sqrt( Zeta(m=12,n=5,s=2) * Zeta(m=12,n=7,s=2) * Zeta(m=12,n=11,s=2) ) as a factor in the formulas. - R. J. Mathar, Feb 04 2021

			0.30231614235706563794776990048019971560241279...

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.
Alessandro Languasco and Pieter Moree, Euler constants from primes in arithmetic progression, arXiv:2406.16547 [math.NT], 2024. See p. 17.
Olivier Ramaré, S. Ettahri, and L. Surel, Fast multi-precision computation of some Euler products, Mathematics of Computation (2021) hal-03381427.

Cf. A003136, A064533, A167135, A301429, A340552.

Digits:= 1000: with(numtheory):
B:= evalf(3^(1/4)*Pi^(1/2)*log(2+sqrt(3))^(1/4)/(2^(5/4)*GAMMA(1/4))):
for t to 500 do p:=ithprime(t): if `or`(`or`(`mod`(p, 12) = 5, `mod`(p, 12) = 7), `mod`(p, 12) = 11) then B:= evalf(B/(1-1/p^2)^(1/2)) end if end do: B;

prec := 200; B = N[(Sqrt[Pi] ((3 Log[2 + Sqrt[3]])/2)^(1/4))/(2 Gamma[1/4]), prec];
For[n = 3, n < 50000, n++, p = Prime[n];
If[Mod[p, 12] != 1, B = B / Sqrt[(1 - 1/p) (1 + 1/p)]]]
Print[B] (* Peter Luschny, Mar 23 2018 *)
(* -------------------------------------------------------------------------- *)
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[(3^(1/4)/2^(5/4)) * Pi^(1/2) * (Log[2 + Sqrt[3]])^(1/4) / Gamma[1/4] * Sqrt[Z[12, 5, 2] * Z[12, 7, 2] * Z[12, 11, 2]], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021 *)

Equals (3^(1/4)/2^(5/4)) * Pi^(1/2) * (log(2 + sqrt(3)))^(1/4) / Gamma(1/4) * Product_{p == 5, 7, 11 (mod 12), p prime} (1 - 1/p^2)^(-1/2).

One can base the definition on p(n) = A167135(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, 3/2, 25/16, 1225/768, 29645/18432, ... -> L. Then A301430 = sqrt(L)*M with M = ((arccosh(2)/6)^(1/4)*Gamma(3/4))/(2*sqrt(Pi)). - Peter Luschny, Mar 29 2018

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, Jan 15 2021

A167134 Primes congruent to {2, 3, 5, 7} mod 11.

Original entry on oeis.org

2, 3, 5, 7, 13, 29, 47, 71, 73, 79, 101, 113, 137, 139, 157, 167, 179, 181, 211, 223, 227, 233, 269, 271, 277, 293, 311, 313, 337, 359, 379, 401, 409, 421, 431, 443, 467, 487, 491, 509, 541, 557, 563, 577, 599, 601, 607, 619, 641, 643, 673, 709, 733, 739, 751

Offset: 1

Klaus Brockhaus, Oct 28 2009

Primes p such that p mod 11 is prime.
Primes of the form 11*n+r where n >= 0 and r is in {2, 3, 5, 7}.
2 and primes congruent to {3, 5, 7, 13} mod 22. - Chai Wah Wu, Apr 29 2025

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A003627, A045326, A003631, A045309, A045314, A042987, A078403, A042993, A167134, A167135, A167119: primes p such that p mod k is prime, for k = 3..13 resp.

[ p: p in PrimesUpTo(760) | p mod 11 in {2, 3, 5, 7} ];
[ p: p in PrimesUpTo(760) | exists(t){ n: n in [0..p div 11] | exists(u){ r: r in {2, 3, 5,7} | p eq (11*n+r) } } ];

Select[Prime[Range[600]],MemberQ[{2, 3, 5, 7},Mod[#,11]]&] (* Vincenzo Librandi, Aug 05 2012 *)

A167119 Primes congruent to 2, 3, 5, 7 or 11 (mod 13).

Original entry on oeis.org

2, 3, 5, 7, 11, 29, 31, 37, 41, 59, 67, 83, 89, 107, 109, 137, 163, 167, 193, 197, 211, 223, 239, 241, 263, 271, 293, 317, 349, 353, 367, 379, 397, 401, 419, 421, 431, 449, 457, 479, 499, 509, 523, 557, 577, 587, 601, 613, 631, 653, 661, 683, 691, 709, 733, 739, 743, 757

Offset: 1

Vladimir Joseph Stephan Orlovsky, Oct 27 2009

Primes which have a remainder mod 13 that is prime.

Union of A141858, A100202, A102732, A140371 and A140373. - R. J. Mathar, Oct 29 2009

			11 mod 13 = 11, 29 mod 13 = 3, 31 mod 13 = 5, hence 11, 29 and 31 are in the sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A003627, A045326, A003631, A045309, A045314, A042987, A078403, A042993, A167134, A167135: primes p such that p mod k is prime, for k = 3..12 resp.

[ p: p in PrimesUpTo(740) | p mod 13 in {2, 3, 5, 7, 11} ]; // Klaus Brockhaus, Oct 28 2009

f[n_]:=PrimeQ[Mod[n,13]]; lst={};Do[p=Prime[n];If[f[p],AppendTo[lst,p]],{n,6,6!}];lst
Select[Prime[Range[4000]],MemberQ[{2, 3, 5, 7, 11},Mod[#,13]]&] (* Vincenzo Librandi, Aug 05 2012 *)

{forprime(p=2, 740, if(isprime(p%13), print1(p, ",")))} \\ Klaus Brockhaus, Oct 28 2009

Edited by Klaus Brockhaus and R. J. Mathar, Oct 28 2009 and Oct 29 2009

A215161 Primes congruent to {2, 3, 5, 7, 11} mod 17.

Original entry on oeis.org

2, 3, 5, 7, 11, 19, 37, 41, 53, 71, 73, 79, 107, 109, 113, 139, 173, 181, 211, 223, 241, 257, 277, 283, 311, 313, 317, 347, 359, 379, 419, 449, 461, 479, 487, 521, 547, 563, 617, 619, 631, 653, 683, 691, 719, 733, 751, 787, 821, 823, 827, 853, 857, 887, 929

Offset: 1

Vincenzo Librandi, Aug 05 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A167119, A167135.

[ p: p in PrimesUpTo(1000) | p mod 17 in {2, 3, 5, 7, 11} ];

Select[Prime[Range[400]],MemberQ[{2,3, 5, 7, 11},Mod[#,17]]&]

A215162 Primes congruent to {2, 3, 5, 7, 11} mod 19.

Original entry on oeis.org

2, 3, 5, 7, 11, 41, 43, 59, 79, 83, 97, 157, 163, 173, 193, 197, 211, 233, 239, 269, 271, 277, 307, 311, 347, 349, 353, 383, 401, 421, 439, 461, 463, 467, 499, 577, 613, 619, 653, 691, 727, 733, 743, 809, 839, 857, 877, 881, 919, 953, 971, 991

Offset: 1

Vincenzo Librandi, Aug 05 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A167119, A167135.

[ p: p in PrimesUpTo(1000) | p mod 19 in {2, 3, 5, 7, 11} ];

Select[Prime[Range[400]],MemberQ[{2,3, 5, 7, 11},Mod[#,19]]&]

cp's OEIS Frontend

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.

Views

Author

Keywords

Examples

Crossrefs

Formula

A301430 Decimal expansion of an analog of the Landau-Ramanujan constant for Loeschian numbers which are sums of two squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A167134 Primes congruent to {2, 3, 5, 7} mod 11.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

A167119 Primes congruent to 2, 3, 5, 7 or 11 (mod 13).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A215161 Primes congruent to {2, 3, 5, 7, 11} mod 17.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

A215162 Primes congruent to {2, 3, 5, 7, 11} mod 19.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica