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%I #13 Feb 26 2020 07:46:28
%S 3,4,8,20,40,230,260,680,1910,2120,6670,9710,10310,23500,25220,37990,
%T 71800
%N Factors k > 2 such that the polynomial x^2 + k*x + 1 produces a new minimum of its Hardy-Littlewood constant.
%C a(18) > 100000.
%C See A331940 for more information on the Hardy-Littlewood constant. The polynomials described by this sequence are increasingly avoiding primes.
%C 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 + 1 for 1 <= 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 3 3.54661 10220078 3.65998
%C 4 1.38342 3982973 1.42637
%C 8 0.91172 2627239 0.94086
%C 20 0.76532 2204290 0.78939
%C ..... ....... ....... .......
%C 25220 0.39947 1151122 0.41224
%C 37990 0.39945 1151126 0.41224
%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, A331945, A331946, A331947, A331948, A331949, A332708.
%K nonn,more
%O 1,1
%A _Hugo Pfoertner_, Feb 20 2020
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