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A331947 Factors k > 1 such that the polynomial k*x^2 - 1 produces a record of its Hardy-Littlewood constant. 6
2, 12, 20, 68, 90, 98, 132, 252, 318, 362, 398, 1722, 259668, 315180, 452042 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

a(16) > 710000.

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 2 <= 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

       2 3.70011  10448345 3.81422

      12 4.15027  11154934 4.27219

      20 4.43326  11753085 4.56136

      68 5.01601  12883801 5.15797

      .. .......  ........ .......

  315180 7.82318  16502584 8.00057

  452042 7.85323  16434699 8.02696

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..15.

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. A221712, A331940, A331941, A331945, A331946, A331948, A331949.

Sequence in context: A266050 A291698 A174478 * A145634 A145610 A185399

Adjacent sequences:  A331944 A331945 A331946 * A331948 A331949 A331950

KEYWORD

nonn,more,hard

AUTHOR

Hugo Pfoertner, Feb 10 2020

STATUS

approved

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Last modified April 20 11:43 EDT 2021. Contains 343135 sequences. (Running on oeis4.)