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A331940 Addends k > 0 such that the polynomial x^2 + x + k produces a record of its Hardy-Littlewood Constant. 8
1, 11, 17, 41, 21377, 27941, 41537, 55661, 115721, 239621, 247757 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

REFERENCES

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.

LINKS

Table of n, a(n) for n=1..11.

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. [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.

PROG

(PARI) \\ 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))

CROSSREFS

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

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

Sequence in context: A187057 A187058 A144051 * A187060 A190800 A191456

Adjacent sequences:  A331937 A331938 A331939 * A331941 A331942 A331943

KEYWORD

nonn,more,hard

AUTHOR

Hugo Pfoertner, Feb 02 2020

STATUS

approved

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Last modified May 15 13:20 EDT 2021. Contains 343920 sequences. (Running on oeis4.)