A331877 -id:A331877 - cp's OEIS frontend

A221712 Hardy-Littlewood constant for x^2+x+41.

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

Offset: 1

N. J. A. Sloane, Jan 26 2013

			3.31977317747142166532355685764988796646855...

Henri Cohen, Number Theory, Vol II: Analytic and Modern Tools, Springer (Graduate Texts in Mathematics 240), 2007.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 265-266.

Christian Axler and Mehdi Hassani, Carleman's inequality over prime numbers, Integers (2021) Vol. 21, #A53.
Karim Belabas and Henri Cohen, Computation of the Hardy-Littlewood constant for quadratic polynomials, PARI/GP script, 2020.
Lea Beneish and Christopher Keyes, How often does a cubic hypersurface have a rational point?, arXiv:2405.06584 [math.NT], 2024. See p. 23.
David Broadhurst, Five families of rapidly convergent evaluations of zeta values, arXiv:2401.08997 [math.NT], 2024.
Henri Cohen, High-precision calculation of Hardy-Littlewood constants, (1998).
Henri Cohen, High precision computation of Hardy-Littlewood constants. [Cached pdf version, with permission]
Stéphane Vinatier and William Reginald Alcorn, Mathematics as seen by an artist: Inspiring mathematical objects, Eur. Math. Soc. Mag. (2025) Vol. 135, 20-31. See p. 26.

Cf. A005846, A056561, A202018, A221713, A319906, A331876, A331877.

\\ See Belabas, Cohen link. Run as HardyLittlewood2(x^2+x+41)/2 after setting the required precision.

More terms from Hugo Pfoertner, Jan 31 2020

A331940 Addends k > 0 such that the polynomial x^2 + x + k produces a record of its Hardy-Littlewood Constant.

Original entry on oeis.org

1, 11, 17, 41, 21377, 27941, 41537, 55661, 115721, 239621, 247757

Offset: 1

Hugo Pfoertner, Feb 02 2020

The Hardy and Littlewood Conjecture F provides an estimate of the number of primes generated by a quadratic polynomial P(x) for 0 <= x <= m in the form C * Integral_{x=2..m} 1/log(x) dx, with C given by an Euler product that is a function of the fundamental discriminant of the polynomial. Cohen describes an efficient method for the computation of C.
The following table provides the record values of C, together with the number of primes np generated by the polynomial x^2 + x + a(n) for x <= 10^8 and the actual ratio 2*np/Integral_{x=2..10^8} 1/log(x) dx.
    a(n)    C       np    C from ratio
       1 2.24147  6456835 2.24110
      11 3.25944  9389795 3.25910
      17 4.17466 12027453 4.17460
      41 6.63955 19132653 6.64073
   21377 6.92868 19962992 6.92894
   27941 7.26400 20931145 7.26497
   41537 7.32220 21092134 7.32085
   55661 7.45791 21483365 7.45664
  115721 7.70935 22210771 7.70912
  239621 7.72932 22268336 7.72909
  247757 8.24741 23762118 8.24757
Jacobson and Williams found significantly larger values of C for very large addends k, e.g. C = 2*5.36708 = 10.73416 for k = 3399714628553118047.

Keith Conrad, Hardy-Littlewood Constants. In: Mathematical Properties of Sequences and Other Combinatorial Structures, eds. Jong-Seon No, Hong-Yeop Song, Tor Helleseth, P. Vijay Kumar, Springer New York, 2003, pages 133-154.

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. [Cached pdf version, with permission]
Keith Conrad, Hardy-Littlewood Constants, (2003).
Michael J. Jacobson Jr. and Hugh C. Williams, New Quadratic Polynomials With High Densities Of Prime Values, Math. Comp., 72, 241, 499-519, 2002.

Cf. A002837, A007635, A014556, A116206, A331877.

Cf. A221712, A221713 (Constants C including factor 1/2).

\\ The function HardyLittlewood2 is provided at the Belabas, Cohen link.
hl2max=0; for(add=0,100,my(hl=HardyLittlewood2(n^2+n+add)); if(hl>hl2max,print1(add,", "); hl2max=hl))

A331876 Number of primes of the form P(k) = k^2 + k + 41 for k <= 10^n, where P(k) is Euler's prime-generating polynomial A202018.

Original entry on oeis.org

2, 11, 87, 582, 4149, 31985, 261081, 2208197, 19132653, 168806741, 1510676803

Offset: 0

Hugo Pfoertner, Jan 30 2020

			a(0) = 2 because 41 and 43 are the 2 primes generated for k <= 1 = 10^0.
a(1) = 11 because 41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151 are the 11 primes generated for k <= 10^1, (A202018(10) = 151).
a(3) = 87 because 87 terms of A202018(0..100) are prime. The 14 composites occur for k = A007634(1..14): 40, 41, 44, 49, 56, ...

Cf. A005846, A007634, A202018, A319906, A331877.

n=0;m=1;for(k=0,10^7,my(j=k^2+k+41);if(isprime(j),n++);if(k==m,m*=10;print1(n,", ")))

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A221712 Hardy-Littlewood constant for x^2+x+41.

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A331940 Addends k > 0 such that the polynomial x^2 + x + k produces a record of its Hardy-Littlewood Constant.

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A331876 Number of primes of the form P(k) = k^2 + k + 41 for k <= 10^n, where P(k) is Euler's prime-generating polynomial A202018.

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