A331946 Factors k > 0 such that k*x^2 + 1 produces a new minimum of its Hardy-Littlewood constant.

1, 5, 11, 17, 29, 41, 89, 101, 461, 521, 761, 941, 1091, 1361, 1889, 2141, 3449, 4289, 5381, 5561, 10709, 15461, 23201, 59309, 70769, 134741, 174929, 329969, 493349

Offset: 1

Author

Hugo Pfoertner, Feb 10 2020

Keywords

• nonn
• more

Comments

a(30) > 600000.

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 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

5 0.66031 1816520 0.67979

11 0.56115 1512897 0.57810

17 0.52244 1392498 0.53816

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

329969 0.20443 430342 0.20883

493349 0.20348 424719 0.20781

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, A331947, A331948.