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%I A332707 #13 Feb 26 2020 07:46:28
%S A332707 3,4,8,20,40,230,260,680,1910,2120,6670,9710,10310,23500,25220,37990,
%T A332707 71800
%N A332707 Factors k > 2 such that the polynomial x^2 + k*x + 1 produces a new minimum of its Hardy-Littlewood constant.
%C A332707 a(18) > 100000.
%C A332707 See A331940 for more information on the Hardy-Littlewood constant. The polynomials described by this sequence are increasingly avoiding primes.
%C A332707 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 A332707    a(n)     C       np    C from ratio
%C A332707       3  3.54661 10220078 3.65998
%C A332707       4  1.38342  3982973 1.42637
%C A332707       8  0.91172  2627239 0.94086
%C A332707      20  0.76532  2204290 0.78939
%C A332707   .....  .......  ....... .......
%C A332707   25220  0.39947  1151122 0.41224
%C A332707   37990  0.39945  1151126 0.41224
%D A332707 Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
%H A332707 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 A332707 Henri Cohen, <a href="/A221712/a221712.pdf">High precision computation of Hardy-Littlewood constants</a>, preprint, 1998. [pdf copy, with permission]
%Y A332707 Cf. A221712, A331940, A331945, A331946, A331947, A331948, A331949, A332708.
%K A332707 nonn,more
%O A332707 1,1
%A A332707 _Hugo Pfoertner_, Feb 20 2020

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