A331948 Nonsquare factors k > 0 such that k*x^2 - 1 produces a new minimum of its Hardy-Littlewood constant.

2, 3, 7, 13, 19, 31, 79, 151, 211, 331, 499, 631, 751, 991, 1171, 2011, 2311, 2671, 3019, 3931, 4159, 4951, 5119, 6451, 7459, 10651, 18379, 32971, 48799, 61051, 78439, 84319, 162451, 199411, 230239, 257371, 404251, 462331, 529699, 584791, 640819

Offset: 1

Author

Hugo Pfoertner, Feb 10 2020

Keywords

• nonn

Comments

a(42) > 10^6.

See A331940 for more information on the Hardy-Littlewood constant. The polynomials described by this sequence are increasingly prime-avoiding.

The following table provides the minimum record values of 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

2 3.70011 10448345 3.81422

3 2.07514 5794128 2.13869

7 0.88360 2411224 0.91046

13 0.87451 2344299 0.89971

.. ....... ....... .......

584791 0.21378 445220 0.21860

640819 0.21229 439946 0.21641

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.