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 A331945 Factors k > 0 such that the polynomial k*x^2 + 1 produces a record of its Hardy-Littlewood constant. 6
 1, 2, 3, 4, 12, 18, 28, 58, 190, 462, 708, 5460, 10602, 39292, 141100, 249582, 288502 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS a(18) > 510000. 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 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.     a(n)    C        np    C from ratio        1 1.37281  3954181  1.41606 (C = A199401)        2 1.42613  4027074  1.47010        3 1.68110  4696044  1.73337        4 2.74563  7605407  2.82915       12 3.36220  9037790  3.46135       .. .......  .......  .......   249582 7.90518 16760196  8.08633   288502 8.21709 17367067  8.40431 REFERENCES Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209. LINKS Karim Belabas, Henri Cohen, Computation of the Hardy-Littlewood constant for quadratic polynomials, PARI/GP script, 2020. Henri Cohen, High precision computation of Hardy-Littlewood constants, preprint, 1998. [pdf copy, with permission] CROSSREFS Cf. A199401, A221712, A331940, A331941, A331946, A331948, A331948, A331949. Sequence in context: A060529 A048073 A002447 * A275067 A295620 A007499 Adjacent sequences:  A331942 A331943 A331944 * A331946 A331947 A331948 KEYWORD nonn,more,hard AUTHOR Hugo Pfoertner, Feb 10 2020 STATUS approved

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Last modified May 12 23:35 EDT 2021. Contains 343829 sequences. (Running on oeis4.)