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

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

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.

a(12) > 3*10^6. - Hugo Pfoertner, Jan 25 2026

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 <= xlim, xlim = {10^8, 10^9, 10^10} and the actual ratios 2*np/Integral_{x=2..xlim} 1/log(x) dx.

a(n)| C | np, x <= | C from ratio, x <= |

| | 10^8 10^9 10^10| 10^8 10^9 10^10|

1 2.24147 6456835 56988601 510007598 2.24110 2.24147 2.24152

11 3.25944 9389795 82872074 741607302 3.25910 3.25952 3.25941

17 4.17466 12027453 106138135 949844417 4.17460 4.17462 4.17463

41 6.63955 19132653 168806741 1510676803 6.64073 6.63950 6.63952

21377 6.92868 19962992 176143136 1576461726 6.92894 6.92806 6.92865

27941 7.26400 20931145 184684776 1652776874 7.26497 7.26402 7.26406

41537 7.32220 21092134 186163235 1666023075 7.32085 7.32217 7.32228

55661 7.45791 21483365 189620557 1696881986 7.45664 7.45815 7.45790

115721 7.70935 22210771 196016006 1754135902 7.70912 7.70970 7.70954

239621 7.72932 22268336 196525660 1758635309 7.72909 7.72974 7.72931

247757 8.24741 23762118 209685056 1876484571 8.24757 8.24733 8.24727

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 (A392688).

From Hermann Stamm-Wilbrandt, Jan 17 2026: (Start)

Enhanced the table above by adding <= columns for 10^9 and 10^10.

For k = 3399714628553118047 from Jacobson and Williams paper "x <= 10^11" is needed (with np = 22066543923) to match C value inclusive 1st digit past decimal point with C from ratio = 10.71693.

For that k (which is not known to be part of this sequence), row as above for comparison:

|10.73416|25153819 251014697 2408242218|8.73062 9.87291 10.58438|

(End)

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 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.

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, A392688.

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