A335963 -id:A335963 - cp's OEIS frontend

A049532 Numbers k such that k^2 + 1 is not squarefree.

7, 18, 32, 38, 41, 43, 57, 68, 70, 82, 93, 99, 107, 117, 118, 132, 143, 157, 168, 182, 193, 207, 218, 232, 239, 243, 251, 257, 268, 282, 293, 307, 318, 327, 332, 343, 357, 368, 378, 382, 393, 407, 408, 418, 432, 437, 443, 457, 468, 482, 493, 500, 507, 515

Offset: 1

Labos Elemer

nonn

The sequence is infinite. For instance, it contains all numbers of the form 7 + 25m. - Emmanuel Vantieghem, Oct 25 2016
More generally, the sequence contains all numbers of the form a(n) + (a(n)^2 + 1) * m for even a(n) and a(n) + (a(n)^2 + 1) * m / 2 for odd a(n). - David A. Corneth, Oct 25 2016
The asymptotic density of this sequence is 1 - A335963 = 0.1051587754... - Amiram Eldar, Jul 08 2020

			a(1) = 7 because 7^2 + 1 = 49 + 1 = 50 is divisible by 25, a square.

R. J. Mathar, Table of n, a(n) for n = 1..7999

Cf. A002522, A059592, A124809, A335963.

[n: n in [1..6*10^2]| not IsSquarefree(n^2+1)]; // Bruno Berselli, Oct 15 2012

n=1;Reap[Do[While[SquareFreeQ[n^2+1],n++];Sow[n];n++,{c,10000}]][[2,1]] (* Zak Seidov, Feb 24 2011 *)

for(n=1,1e4,if(!issquarefree(n^2+1),print1(n", "))) \\ Charles R Greathouse IV, Feb 24 2011

A059592(a(n)) > 1; A124809(n) = a(n)^2 + 1. - Reinhard Zumkeller, Nov 08 2006

Definition rewritten by Bruno Berselli, Oct 15 2012

Mathematica updated by Jean-François Alcover, Jun 19 2013

A340004 Decimal expansion of Product_{primes p == 1 (mod 5)} p^2/(p^2-1).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jan 15 2021

This constant is called Euler product 2==1 modulo 5 (see Mathar's Definition 5 formula (38)) or equivalently zeta 2==1 modulo 5.

			1.01091516060101952260495658428951492...

Vaclav Kotesovec, Table of n, a(n) for n = 1..501
Salma Ettahri, Olivier Ramaré, and Léon Surel, Fast multi-precision computation of some Euler products, arXiv:1908.06808 [math.NT], 2019 p.20 (100 digits precision data).
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2014-2015, Section 3.3. zeta_{5,1}(2).

Cf. A340127, A340628, A340629, A340665, A340794, A340839.
Cf. A175646, A301429, A333240.
Cf. A175647, A248930, A248938, A335963.
Cf. A004615, A340576, A340577, A340578, A334826.

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[5, 1, 2], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021, took 20 minutes *)

Equals Sum_{k>=1} 1/A004615(k)^2. - Amiram Eldar, Jan 24 2021

Equals exp(-gamma/2)*Pi/(A340839^2*sqrt(5*log((1 + sqrt (5))/2))). - Artur Jasinski, Jan 30 2021

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

A340628 Decimal expansion of Product_{primes p == 4 (mod 5)} (p^2+1)/(p^2-1).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jan 13 2021

			1.009935934829401027349038488241778167715858547548801013...

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 products, arXiv:1908.06808 [math.NT], 2019 p.20 (70 digits precision data).
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, On the constant in the Mertens product for arithmetic progressions. I. Identities, Funct. Approx. Comment. Math. Volume 42, Number 1 (2010), 17-27, (preliminary version).
Richard J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Cf. A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340577, A340578, A340629.

evalf(Re(2*Pi^2/(5*sqrt(13*((I*Pi^2*(1/150)-I*polylog(2, (-1)^(2/5)))^2+((1/150)*(11*I)*Pi^2+I*polylog(2, (-1)^(4/5)))^2)))), 120) # Vaclav Kotesovec, Jan 20 2021, after formula by Pascal Sebah

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; PrintTemporary["iteration = ", w, ", difference = ", N[difz, digits]]; w++]; Exp[sumz]);
$MaxExtraPrecision = 1000; digits = 121; Chop[N[1/(Z[5,4,4]/Z[5,4,2]^2), digits]] (* Vaclav Kotesovec, Jan 15 2021, took over 20 minutes *)
digits = 121; digitize[c_] := RealDigits[Chop[N[c, digits]], 10, digits][[1]];
cl[x_] :=I(PolyLog[2,(-1)^x] - PolyLog[2,-(-1)^(1-x)]);
A340628 :=(4 Pi^2)/(5 Sqrt[13])/ Sqrt[cl[2/5]^2 + cl[4/5]^2];
digitize[A340628] (* Peter Luschny, Jan 23 2021 *)

Equals 6*sqrt(5)/(13*A340629).
Equals 6*sqrt(13)*Pi^2/(195*g) where g = sqrt(Cl2(2*Pi/5)^2 + Cl2(4*Pi/5)^2) = 1.0841621352693895..., and Cl2 is the Clausen function of order 2. Formula by Pascal Sebah (personal communication). - Artur Jasinski, Jan 20 2021
Equals A340127^2/A340809. - R. J. Mathar, Jan 22 2021
Equals Sum_{q in A004618} 2^A001221(q)/q^2. - R. J. Mathar, Jan 27 2021

Corrected and more terms from Vaclav Kotesovec, Jan 15 2021

A340576 Decimal expansion of Product_{primes p == 5 (mod 6)} 1/(1-1/p^2).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jan 12 2021

The four similar sequences for products of primes mod 6 are these:
  A175646 for Product_{primes p == 1 (mod 6)} 1/(1-1/p^2),
  A340576 for Product_{primes p == 5 (mod 6)} 1/(1-1/p^2),
  A340577 for Product_{primes p == 1 (mod 6)} 1/(1+1/p^2),
  A340578 for Product_{primes p == 5 (mod 6)} 1/(1+1/p^2).

			1.06054829316911072817412636430987203493077130204487163127994372...

R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, Zeta_{6,5}(2) in section 3.2.

Similar constants: A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340577, A340578.

Cf. A259548.

a := n -> 3^(2^(-n-2))*((1-3^(-2^(n+1)))/2)^(2^(-n-1)):
b := n -> Zeta(n)/Im(polylog(n, (-1)^(2/3))):
c := n -> a(n)*b(2^(n+1))^(1/2^(n+1)):
Digits := 107: evalf((3/4)*mul(c(n), n=0..9)); # Peter Luschny, Jan 14 2021

digits = 105;
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];
pB = (3/4)*Product[gv[2^n*2]^(2^-(n+1)), {n, 0, 11}] // N[#, precision]&;
RealDigits[pB, 10, digits][[1]] (* Most of this code is 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[Z[6,5,2], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021 *)

g = A143298 = (9 - PolyGamma(1, 2/3) + PolyGamma(1, 4/3))/(4 sqrt(3));
h = A301429;
Equals (3*sqrt(3)*h^2)/2.
Equals (3/4)*A333240.
A340577 = Pi^4/(243*g*h^2);
A340578 = (45*g*h^2)/(2*Pi^2).
Equals Pi^2/(9*A175646). - Artur Jasinski, Jan 11 2021
Equals Sum_{k>=1} 1/A259548(k)^2. - Amiram Eldar, Jan 24 2021

A340577 Decimal expansion of Product_{primes p == 1 (mod 6)} 1/(1+1/p^2).

Original entry on oeis.org

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, 1

Offset: 0

Jean-François Alcover, Jan 12 2021

			0.96755040251956188660947077043906773001524912960304386356302398...

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

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

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

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

A340578 Decimal expansion of Product_{primes p == 5 (mod 6)} 1/(1+1/p^2).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jan 12 2021

			0.94450093450470098673429109419144434254611078086906676955735771...

Similar constants: A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340577.

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]&;
pD = (45*pB*Lv[2])/(4*Pi^2);
RealDigits[pD, 10, digits][[1]] (* Most of this code is 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[Z[6, 5, 4]/Z[6, 5, 2], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021 *)

A340629 Decimal expansion of Product_{primes p == 1 (mod 5)} (p^2+1)/(p^2-1).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jan 13 2021

			1.0218780604187566757444489146002708261704607377325...

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 products, arXiv:1908.06808 [math.NT], 2019, p. 20 (70-digit-precision data).
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, On the constant in the Mertens product for arithmetic progressions. I. Identities, Funct. Approx. Comment. Math. Volume 42, Number 1 (2010), 17-27, (preliminary version).
Richard J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.

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

evalf(Re(15*sqrt((1/13)*(5*((I*Pi^2*(1/150)-I*polylog(2, (-1)^(2/5)))^2+((1/150)*(11*I)*Pi^2+I*polylog(2, (-1)^(4/5)))^2)))/Pi^2), 120) # Vaclav Kotesovec, Jan 20 2021, after formula by Pascal Sebah.

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; PrintTemporary["iteration = ", w, ", difference = ", N[difz, digits]]; w++]; Exp[sumz]);
$MaxExtraPrecision = 1000; digits = 121; Chop[N[1/(Z[5,1,4]/Z[5,1,2]^2), digits]] (* Vaclav Kotesovec, Jan 15 2021, took over 20 minutes *)
digits = 121; digitize[c_] := RealDigits[Chop[N[c, digits]], 10, digits][[1]];
cl[x_] := I (PolyLog[2, (-1)^x] - PolyLog[2, -(-1)^(1 - x)]);
A340629 := (15 Sqrt[65]/(26 Pi^2)) Sqrt[cl[2/5]^2 + cl[4/5]^2];
digitize[A340629] (* Peter Luschny, Jan 23 2021 *)

Equals 6*sqrt(5)/(13*A340628).
Equals A340004^2/A340808. - R. J. Mathar, Jan 15 2021
Equals 15*sqrt(65)*g/(13*Pi^2) where g = sqrt(Cl2(2*Pi/5)^2 + Cl2(4*Pi/5)^2) = 1.0841621352693895..., and Cl2 is the Clausen function of order 2. Formula by Pascal Sebah (personal communication). - Artur Jasinski, Jan 20 2021
Equals Sum_{q in A004615} 2^A001221(q)/q^2. - R. J. Mathar, Jan 27 2021

Corrected and more terms from Vaclav Kotesovec, Jan 15 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

A049533 Numbers k such that k^2+1 is squarefree.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 39, 40, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Labos Elemer

nonn

Estermann proved that a(n) ~ kn with k = 1.117...; more precisely, there are cx + O(x^(2/3) log x) terms up to x, where c = 1/k = Product (1 - 2/p^2) where the product is over primes p which are 1 mod 4. Heath-Brown improves the error term to O(x^(7/12) log x). - Charles R Greathouse IV, Oct 16 2017, corrected by Amiram Eldar, Jul 08 2020

There are 89489 terms up to 10^5, 894856 terms up to 10^6, 8948417 up to 10^7, 89484102 up to 10^8, and 894841314 up to 10^9. - Charles R Greathouse IV, Nov 26 2017, corrected and extended by Amiram Eldar, Jul 08 2020

			10 is a member because 10^2 + 1 = 100 + 1 = 101 is squarefree.
Reasons why certain numbers are excluded: 7^2+1 = 2*5^2, 18^2+1 = 13*5^2, 32^2+1 = 41*5^2, 38^2+1 = 5*17^2, 41^2+1 = 2*29^2, 43^2+1 = 74*5^2, 57^2+1 = 130*5^2, 82^2+1 = 269*5^2. - Neven Juric, Oct 06 2008

Michael De Vlieger, Table of n, a(n) for n = 1..10000
T. Estermann, Einige Sätze über quadratfreie Zahlen, Math. Ann. 105 (1931), pp. 653-662.
D. R. Heath-Brown, Square-free values of n^2 + 1, Acta Arithmetica 155:1 (2012), pp. 1-13; arXiv:1010.6217 [math.NT], 2010-2012.

Complement of A049532.

Cf. A059592, A069987, A335963.

[ n: n in [1..100] | IsSquarefree(n^2+1) ]; // Vincenzo Librandi, Dec 25 2010

Select[Range@ 80, SquareFreeQ[#^2 + 1] &] (* Michael De Vlieger, Aug 09 2017 *)

isok(n) = issquarefree(n^2+1); \\ Michel Marcus, Feb 09 2016

Numbers k such that A059592(k) = 1. - Reinhard Zumkeller, Nov 08 2006

cp's OEIS Frontend

A049532 Numbers k such that k^2 + 1 is not squarefree.

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A340004 Decimal expansion of Product_{primes p == 1 (mod 5)} p^2/(p^2-1).

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

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A340628 Decimal expansion of Product_{primes p == 4 (mod 5)} (p^2+1)/(p^2-1).

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A340576 Decimal expansion of Product_{primes p == 5 (mod 6)} 1/(1-1/p^2).

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A340577 Decimal expansion of Product_{primes p == 1 (mod 6)} 1/(1+1/p^2).

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A340578 Decimal expansion of Product_{primes p == 5 (mod 6)} 1/(1+1/p^2).

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A340629 Decimal expansion of Product_{primes p == 1 (mod 5)} (p^2+1)/(p^2-1).

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