A332708 Factors k >= 0 such that the polynomial x^2 + k*x + 1 produces a record of its Hardy-Littlewood constant.

1, 3, 21, 231, 879, 1011, 1089, 1659, 2751

Offset: 1

Author

Hugo Pfoertner, Feb 20 2020

Keywords

• nonn
• more

Comments

a(10) > 80000.

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) = x^2 + a(n)*x + 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 2.24147 6456835 2.31230

3 3.54661 10220078 3.65998

21 5.58679 16096923 5.76458

231 5.74156 16543757 5.92460

879 5.83722 16813676 6.02126

1011 5.92725 17073610 6.11435

1089 6.03701 17392675 6.22861

1659 6.04359 17413761 6.23617

2751 7.46622 21508374 7.70252

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, A331945, A331946, A331947, A331948, A331949, A332707.