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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

Table of n, a(n) for n=1..17.

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.)