|
%I #12 Feb 21 2020 10:59:32
%S 1,2,3,4,12,18,28,58,190,462,708,5460,10602,39292,141100,249582,288502
%N Factors k > 0 such that the polynomial k*x^2 + 1 produces a record of its Hardy-Littlewood constant.
%C a(18) > 510000.
%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 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 1 1.37281 3954181 1.41606 (C = A199401)
%C 2 1.42613 4027074 1.47010
%C 3 1.68110 4696044 1.73337
%C 4 2.74563 7605407 2.82915
%C 12 3.36220 9037790 3.46135
%C .. ....... ....... .......
%C 249582 7.90518 16760196 8.08633
%C 288502 8.21709 17367067 8.40431
%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. A199401, A221712, A331940, A331941, A331946, A331948, A331948, A331949.
%K nonn,more,hard
%O 1,2
%A _Hugo Pfoertner_, Feb 10 2020
| 