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 A331947 Factors k > 1 such that the polynomial k*x^2 - 1 produces a record of its Hardy-Littlewood constant. 6

%I

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

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

%C a(16) > 710000.

%C See A331940 for more information on the Hardy-Littlewood constant. The polynomials described by this sequence have an increasing rate of generating primes.

%C 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.

%C a(n) C np C from ratio

%C 2 3.70011 10448345 3.81422

%C 12 4.15027 11154934 4.27219

%C 20 4.43326 11753085 4.56136

%C 68 5.01601 12883801 5.15797

%C .. ....... ........ .......

%C 315180 7.82318 16502584 8.00057

%C 452042 7.85323 16434699 8.02696

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

%H Karim Belabas, Henri Cohen, <a href="/A221712/a221712.gp.txt">Computation of the Hardy-Littlewood constant for quadratic polynomials</a>, PARI/GP script, 2020.

%H Henri Cohen, <a href="/A221712/a221712.pdf">High precision computation of Hardy-Littlewood constants</a>, preprint, 1998. [pdf copy, with permission]

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

%K nonn,more,hard

%O 1,1

%A _Hugo Pfoertner_, Feb 10 2020

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