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3, 4, 8, 20, 40, 230, 260, 680, 1910, 2120, 6670, 9710, 10310, 23500, 25220, 37990, 71800

a(1718) > 60000100000.

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Factors k > 2 such that the polynomial x^2 + k*x + 1 produces a new minimum of its Hardy-Littlewood Constantconstant.

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

The following table provides the minimum 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^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.

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 minimum 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

3 3.54661 10220078 3.65998

4 1.38342 3982973 1.42637

8 0.91172 2627239 0.94086

20 0.76532 2204290 0.78939

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

25220 0.39947 1151122 0.41224

37990 0.39945 1151126 0.41224

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

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

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