A331941 -id:A331941 - cp's OEIS frontend

A206709 Number of primes of the form b^2 + 1 for b <= 10^n.

5, 19, 112, 841, 6656, 54110, 456362, 3954181, 34900213, 312357934, 2826683630, 25814570672, 237542444180, 2199894223892

Offset: 1

Michel Lagneau, Feb 13 2012

Conjecture: The number of primes of the form b^2 + 1 and less than n is asymptotic to 3*n/(4*log(n)).
Examples:
n = 10^3, a(n) = 112 and 3*10^3/(4*log(10^3)) = 108.573...;
n = 10^4, a(n) = 841 and 3*10^4/(4*log(10^4)) = 814.302...;
n = 10^10, a(n) = 312357934 and 3*10^10/(4*log(10^10)) = 325720861.42...
a(n) = A083844(2*n), but not always! The only known exception to this rule is at n = 1. - Arkadiusz Wesolowski, Jul 21 2012
From Jacques Tramu, Sep 14 2018: (Start)
In the table below, K = 0.686413 and pi(10^n) = A000720(10^n):
.
   n         a(n)   K*pi(10^n)
  ==  ===========  ===========
   1            5            3
   2           19           17
   3          112          115
   4          841          843
   5         6656         6584
   6        54110        53882
   7       456362       456175
   8      3954181      3954737
   9     34900213     34902408
  10    312357934    312353959
  11   2826683630   2826686358
  12  25814570672  25814559712
(End)
For a comparison with the estimate that results from the Hardy and Littlewood Conjecture F, see A331942. - Hugo Pfoertner, Feb 03 2020

			a(2) = 19 because there are 19 primes of the form b^2 + 1 for b less than 10^2: 2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601, 2917, 3137, 4357, 5477, 7057, 8101 and 8837.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 264.

Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, and Stephan Ramon Garcia, The Bateman-Horn Conjecture: Heuristics, History, and Applications, arXiv:1807.08899 [math.NT], 2018-2019. See Table 2. p. 8.
Marek Wolf, Search for primes of the form m^2+1, arXiv:0803.1456 [math.NT], 2008-2010.

Cf. A002496, A083844, A215047, A331941, A331942.

for n from 1 to 9 do : i:=0:for m from 1 to 10^n do:x:=m^2+1:if type(x,prime)=true then i:=i+1:else fi:od: printf ( "%d %d \n",n,i):od:

1 + Accumulate@ Array[Count[Range[10^(# - 1) + 1, 10^#], ?(PrimeQ[#^2 + 1] &)] &, 7] (* _Michael De Vlieger, Sep 18 2018 *)

a(n)=sum(n=1,10^n,ispseudoprime(n^2+1)) \\ Charles R Greathouse IV, Feb 13 2012

from sympy import isprime
def A206709(n):
    c, b, b2, n10 = 0, 1, 2, 10**n
    while b <= n10:
        if isprime(b2):
            c += 1
        b += 1
        b2 += 2*b - 1
    return c # Chai Wah Wu, Sep 17 2018

a(11)-a(12) from Arkadiusz Wesolowski, Jul 21 2012

a(13)-a(14) from Jinyuan Wang, Feb 24 2020

A199401 Decimal expansion of constant Product_{p>=3} (1 - (-1)^((p-1)/2)/(p-1)). Hardy-Littlewood constant of x^2 + 1.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Nov 05 2011

Arises in studying A002496.

The constant is Product_{primes p} (1-chi(p)/(p-1)) where chi is the Dirichlet character A101455. Its Euler expansion is (1/(L(m=4,r=2,s=1)* zeta(m=4,n=3,s=2)) *Product_{s>=2} zeta(m=4,n=1,s)^gamma(s), where L and zeta are the functions tabulated in arXiv:1008.2547 and gamma is the sequence A001037. In particular L(m=4,r=2,s=1) = A003881 and zeta(m=4,n=1,s=2)=A175647. - R. J. Mathar, Nov 29 2011

			1.372813462818246009112192696727...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.1, p. 85.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 264.

T. Amdeberhan, L. A. Median, and V. H. Moll, Arithmetical properties of a sequence arising from an arctangent sum, J. Numb. Theory 128 (2008) 1807-1846, eq. (1.10).
Karim Belabas and Henri Cohen, Computation of the Hardy-Littlewood constant for quadratic polynomials, PARI/GP script, 2020.
Henri Cohen, High-precision computation of Hardy-Littlewood constants, (1998). [pdf copy, with permission]
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70. See Section 5.41.
Richard J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.
Marek Wolf, Search for primes of the form m^2+1, arXiv:0803.1456 [math.NT], 2008-2010.

Cf. A002496.
Cf. A001037, A003881, A083844, A175647, A221712, A331942.
Equals 2*constant given by A331941.

\\ See Belabas, Cohen link. Run as HardyLittlewood2(x^2+1) after setting the required precision.

Extended title, a(30) and beyond from Hugo Pfoertner, Feb 16 2020

A331945 Factors k > 0 such that the polynomial k*x^2 + 1 produces a record of its Hardy-Littlewood constant.

Original entry on oeis.org

1, 2, 3, 4, 12, 18, 28, 58, 190, 462, 708, 5460, 10602, 39292, 141100, 249582, 288502

Offset: 1

Hugo Pfoertner, Feb 10 2020

a(18) > 510000.
See A331940 for more information on the Hardy-Littlewood constant. The polynomials described by this sequence have an increasing rate of generating primes.
The following table provides the record values of the Hardy-Littlewood constant C, together with the number of primes np generated by the polynomial P(x) = a(n)*x^2 + 1 for 1 <= x <= r = 10^8 and the actual ratio np*(P(r)/r)/Integral_{x=2..P(r)} 1/log(x) dx.
    a(n)    C        np    C from ratio
       1 1.37281  3954181  1.41606 (C = A199401)
       2 1.42613  4027074  1.47010
       3 1.68110  4696044  1.73337
       4 2.74563  7605407  2.82915
      12 3.36220  9037790  3.46135
      .. .......  .......  .......
  249582 7.90518 16760196  8.08633
  288502 8.21709 17367067  8.40431

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.

Karim Belabas, Henri Cohen, Computation of the Hardy-Littlewood constant for quadratic polynomials, PARI/GP script, 2020.
Henri Cohen, High precision computation of Hardy-Littlewood constants, preprint, 1998. [pdf copy, with permission]

Cf. A199401, A221712, A331940, A331941, A331946, A331948, A331948, A331949.

A331947 Factors k > 1 such that the polynomial k*x^2 - 1 produces a record of its Hardy-Littlewood constant.

Original entry on oeis.org

2, 12, 20, 68, 90, 98, 132, 252, 318, 362, 398, 1722, 259668, 315180, 452042

Offset: 1

Hugo Pfoertner, Feb 10 2020

a(16) > 710000.
See A331940 for more information on the Hardy-Littlewood constant. The polynomials described by this sequence have an increasing rate of generating primes.
The following table provides the record values of the Hardy-Littlewood constant C, together with the number of primes np generated by the polynomial P(x) = a(n)*x^2 - 1 for 2 <= x <= r = 10^8 and the actual ratio np*(P(r)/r)/Integral_{x=2..P(r)} 1/log(x) dx.
    a(n)    C        np    C from ratio
       2 3.70011  10448345 3.81422
      12 4.15027  11154934 4.27219
      20 4.43326  11753085 4.56136
      68 5.01601  12883801 5.15797
      .. .......  ........ .......
  315180 7.82318  16502584 8.00057
  452042 7.85323  16434699 8.02696

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.

Karim Belabas, Henri Cohen, Computation of the Hardy-Littlewood constant for quadratic polynomials, PARI/GP script, 2020.
Henri Cohen, High precision computation of Hardy-Littlewood constants, preprint, 1998. [pdf copy, with permission]

Cf. A221712, A331940, A331941, A331945, A331946, A331948, A331949.

A331949 Addends k > 0 such that x^2 + k produces a new minimum of its Hardy-Littlewood Constant.

Original entry on oeis.org

1, 2, 5, 11, 14, 26, 41, 89, 101, 194, 314, 341, 446, 689, 1091, 1154, 1889, 2141, 3449, 3506, 5561, 6254, 8126, 8774, 10709, 13166, 15461, 23201, 24569, 30014, 81626, 162686

Offset: 1

Hugo Pfoertner, Feb 04 2020

This sequence is almost identical to A003420. However, there is an additional term 446 and after 30014 the number 81626 follows, while in A003420, 81149 is present between 30014 and 81626. With
C(m) = Product_{p=primes} 1 - Kronecker(-4*m,p)/(p - 1) (Hardy-Littlewood)
L1(m) = Sum_{j>0} Kronecker(-4*m,j)/j (L-function of the Dirichlet series)
the following table shows the differences:
              Criterion
          decrease increase
      k      C        L1
     341  0.28309  2.38177
     446  0.28272  2.38014 not in A003420 because L1(446) < L1(341)
     689  0.28193  2.39370
   ...
   30014  0.21541  3.08274
   81149  0.21560  3.08792 not in this sequence because C(81149) > C(30014)
   81626  0.20883  3.17785
  162686  0.20478  3.24017

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.

Karim Belabas, Henri Cohen, Computation of the Hardy-Littlewood constant for quadratic polynomials, PARI/GP script, 2020.
Henri Cohen, High-precision computation of Hardy-Littlewood constants, preprint, 1998. [pdf copy, with permission]
D. Shanks, Systematic examination of Littlewood's bounds on L(1,chi), Proc. Sympos. Pure Math., 24 (1973). Amer. Math. Soc. (Annotated scanned copy)

Cf. A003420, A003521, A331940, A331941.

\\ The function HardyLittlewood2 is provided at the Belabas, Cohen link.
hl2min=oo; for(add=1,500,my(hl=HardyLittlewood2(n^2+add));if(hl

A331942 a(n) = number of primes of the form P(k) = k^2 + 1 <= 10^n as predicted by the Hardy and Littlewood Conjecture F, rounded to nearest integer. The actual number of primes is A083844(n).

Original entry on oeis.org

1, 4, 9, 20, 48, 121, 317, 855, 2356, 6609, 18787, 53970, 156385, 456404, 1340088, 3955219, 11726332, 34903256, 104251560, 312353236, 938461459, 2826668497, 8533343468, 25814350227, 78239112814, 237541788793, 722354115787, 2199893807666, 6708847354653, 20485514756657

Offset: 1

Hugo Pfoertner, Feb 02 2020

nonn

Comparison of actual and approximated number of primes < 10^n:
   Limit
   10^n
    | A083844(n)
    |   | a(n)
    |   | |   (a(n) - A083844(n))/A083844(n)
   10^1 2 1  -0.50000
   10^2 4 4   0.0
   10^3 10 9  -0.10000
   10^4 19 20  0.052632
   10^5 51 48  -0.058824
   10^6 112 121  0.080357
   10^7 316 317  0.0031646
   10^8 841 855  0.016647
   10^9 2378 2356  -0.0092515
  10^10 6656 6609  -0.0070613
  10^11 18822 18787  -0.0018595
  10^12 54110 53970  -0.0025873
  10^13 156081 156385  0.0019477
  10^14 456362 456404  9.2032E-5
  10^15 1339875 1340088  0.00015897
  10^16 3954181 3955219  0.00026251
  10^17 11726896 11726332  -4.8095E-5
  10^18 34900213 34903256   8.7191E-5
  10^19 104248948 104251560   2.5055E-5
  10^20 312357934 312353236  -1.5040E-5
  10^21 938457801 938461459   3.8979E-6
  10^22 2826683630 2826668497  -5.3536E-6
  10^23 8533327397 8533343468   1.8833E-6
  10^24 25814570672 25814350227  -8.5396E-6
  10^25 78239402726 78239112814  -3.7054E-6
  10^26 237542444180 237541788793  -2.7590E-6
  10^27 722354138859 722354115787  -3.1940E-8
  10^28 2199894223892 2199893807666  -1.8920E-7

Keith Conrad, Hardy-Littlewood Constants, (2003).

Cf. A083844, A331941.

C=0.68640673140912300455609634836350943408916655062787977896811707366392;
x=1.0;S10=sqrt(10);for(k=1,30,x*=s10;print1(round(C*intnum(y=2,x,1/log(y))),", "))

b(m) = round (C * Integral_{x=2..m} 1/log(x) dx), with C ~= 0.6864067314..., the Hardy-Littlewood constant for k^2 + 1 (A331941); a(n) = b(10^(n/2)).

A337606 Decimal expansion of the Gaussian twin prime constant: the Hardy-Littlewood constant for A096012.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Sep 04 2020

The name of this constant was suggested by Finch (2003).
Gaussian twin primes on the line x + i in the complex plane are Gaussian primes pair of the form (m - 1 + i, m + 1 + i). The numbers m are numbers such that (m-1)^2 + 1 and (m+1)^2 + 1 are both primes (A096012 plus 1).
Shanks (1960) conjectured that the number of these pairs with m <= x is asymptotic to c * li_2(x), where li_2(x) = Integral_{t=2..n} (1/log(t)^2) dt, and c is this constant. He defined c as in the formula section and evaluated it by 0.4876.
The first 100 digits of 4*c were calculated by Ettahri et al. (2019).

			0.487622778111571768611646391452388423131677124429735...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 90.

Keith Conrad, Hardy-Littlewood constants in: Mathematical properties of sequences and other combinatorial structures, Jong-Seon No et al. (eds.), Kluwer, Boston/Dordrecht/London, 2003, pp. 133-154, alternative link.
Salma Ettahri, Olivier Ramaré, Léon Surel, Fast multi-precision computation of some Euler products, arXiv:1908.06808 [math.NT], 2019 (Section 8).
Daniel Shanks, , A note on Gaussian twin primes, Mathematics of Computation, Vol. 14, No. 70 (1960), pp. 201-203.

Cf. A002496, A005574, A096012, A206328.

Similar constants: A005597, A331941, A337607, A337608.

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]);
Zs[m_, n_, s_] := (w = 2; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = (s^w - s) * P[m, n, w]/w; sumz = sumz + difz; w++]; Exp[-sumz]);
$MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Pi^2/8 * Zs[4, 1, 4]/Z[4, 1, 2]^2, digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021 *)

Equals (Pi^2/8) * Product_{primes p == 1 (mod 4)} (1 - 4/p)*((p + 1)/(p - 1))^2.

More digits from Vaclav Kotesovec, Jan 15 2021

A337607 Decimal expansion of Shanks's constant: the Hardy-Littlewood constant for A000068.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Sep 04 2020

Named by Finch (2003) after the American mathematician Daniel Shanks (1917 - 1996).
Shanks (1961) conjectured that the number of primes of the form m^4 + 1 (A037896) with m <= x is asymptotic to c * li(x), where li(x) is the logarithmic integral function and c is this constant. He defined c as in the formula section and evaluated it by 0.66974.
The first 100 digits of this constant were calculated by Ettahri et al. (2019).

			0.669740969937071220538922431571764406688370157436482...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 90.

Keith Conrad, Hardy-Littlewood constants in: Mathematical properties of sequences and other combinatorial structures, Jong-Seon No et al. (eds.), Kluwer, Boston/Dordrecht/London, 2003, pp. 133-154, alternative link.
Salma Ettahri, Olivier Ramaré, Léon Surel, Fast multi-precision computation of some Euler products, arXiv:1908.06808 [math.NT], 2019 (Corollary 1.8).
Mohan Lal, Primes of the form n^4 + 1, Mathematics of Computation, Vol. 21, No. 98 (1967), pp. 245-247.
Daniel Shanks, On Numbers of the form n^4 + 1, Mathematics of Computation, Vol. 15, No. 74 (1961), pp. 186-189.
Daniel Shanks, Lal's constant and generalizations, Mathematics of Computation, Vol. 21, No. 100 (1967), pp. 705-707.
Eric Weisstein's World of Mathematics, Lal's Constant.

Cf. A000068, A037896, A088367, A334826.

Similar constants: A005597, A331941, A337606, A337608.

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]);
Zs[m_, n_, s_] := (w = 2; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = (s^w - s) * P[m, n, w]/w; sumz = sumz + difz; w++]; Exp[-sumz]);
$MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Pi^2/(16*Log[1+Sqrt[2]]) * Zs[8, 1, 4]/Z[8, 1, 2]^2, digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021 *)

Equals (Pi^2/(16*log(1+sqrt(2)))) * Product_{primes p == 1 (mod 8)} (1 - 4/p)*((p + 1)/(p - 1))^2 = (Pi/8) * A088367 * A334826.

More digits from Vaclav Kotesovec, Jan 15 2021

A337608 Decimal expansion of Lal's constant: the Hardy-Littlewood constant for A217795.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Sep 04 2020

Shanks (1967) conjectured that the number of primes of the form (m + 1)^4 + 1 such that (m - 1)^4 + 1 is also a prime (A217795 plus 1), with m <= x, is asymptotic to c * li_2(x), where li_2(x) = Integral_{t=2..n} (1/log(t)^2) dt, and c is this constant. He defined c as in the formula section, evaluated it by 0.79220 and named it after the mathematician Mohan Lal, who conjectured the asymptotic formula without evaluating this constant.

The first 100 digits of this constant were calculated by Ettahri et al. (2019).

			0.792208238167541668775455566579024101128932250986221...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, pp. 90-91.

Keith Conrad, Hardy-Littlewood constants in: Mathematical properties of sequences and other combinatorial structures, Jong-Seon No et al. (eds.), Kluwer, Boston/Dordrecht/London, 2003, pp. 133-154, alternative link.
Salma Ettahri, Olivier Ramaré, Léon Surel, Fast multi-precision computation of some Euler products, arXiv:1908.06808 [math.NT], 2019 (Corollary 1.9).
Mohan Lal, Primes of the form n^4 + 1, Mathematics of Computation, Vol. 21, No. 98 (1967), pp. 245-247.
Daniel Shanks, Lal's constant and generalizations, Mathematics of Computation, Vol. 21, No. 100 (1967), pp. 705-707.
Eric Weisstein's World of Mathematics, Lal's Constant.

Cf. A000068, A037896, A088367, A210630, A217795, A217796.

Similar constants: A005597, A331941, A337606, A337607.

$MaxExtraPrecision = 1000; digits = 121;
f[p_] := (p-8)*(p+1)^4/((p-1)^4*p);
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[8, 1, m] - 1/17^m); sump = sump + difp; m++];
RealDigits[Chop[N[f[17] * Pi^4/(2^7 * Log[1+Sqrt[2]]^2) * Exp[sump], digits]], 10, digits - 1][[1]] (* Vaclav Kotesovec, Jan 16 2021 *)

Equals (Pi^4/(2^7 * log(1+sqrt(2))^2)) * Product_{primes p == 1 (mod 8)} (1 - 4/p)^2 * ((p + 1)/(p - 1))^4 * p*(p-8)/(p-4)^2 = (Pi^2/32) * A088367^2 * A334826^2 * A210630 = 2 * A337607^2 * A210630.

More terms from Vaclav Kotesovec, Jan 16 2021

cp's OEIS Frontend

A206709 Number of primes of the form b^2 + 1 for b <= 10^n.

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A199401 Decimal expansion of constant Product_{p>=3} (1 - (-1)^((p-1)/2)/(p-1)). Hardy-Littlewood constant of x^2 + 1.

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A331945 Factors k > 0 such that the polynomial k*x^2 + 1 produces a record of its Hardy-Littlewood constant.

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A331947 Factors k > 1 such that the polynomial k*x^2 - 1 produces a record of its Hardy-Littlewood constant.

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A331949 Addends k > 0 such that x^2 + k produces a new minimum of its Hardy-Littlewood Constant.

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A331942 a(n) = number of primes of the form P(k) = k^2 + 1 <= 10^n as predicted by the Hardy and Littlewood Conjecture F, rounded to nearest integer. The actual number of primes is A083844(n).

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A337606 Decimal expansion of the Gaussian twin prime constant: the Hardy-Littlewood constant for A096012.

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A337607 Decimal expansion of Shanks's constant: the Hardy-Littlewood constant for A000068.

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