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

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

Offset: 1

Author

Hugo Pfoertner, Feb 10 2020

Keywords

• nonn
• more
• hard

Comments

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

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