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

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

Offset: 1

Author

Hugo Pfoertner, Feb 10 2020

Keywords

• nonn
• more
• hard

Comments

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

References

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

Links

• 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]

Crossrefs

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