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A331948 Nonsquare factors k > 0 such that k*x^2 - 1 produces a new minimum of its Hardy-Littlewood constant. 5
2, 3, 7, 13, 19, 31, 79, 151, 211, 331, 499, 631, 751, 991, 1171, 2011, 2311, 2671, 3019, 3931, 4159, 4951, 5119, 6451, 7459, 10651, 18379, 32971, 48799, 61051, 78439, 84319, 162451, 199411, 230239, 257371, 404251, 462331, 529699, 584791, 640819 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

a(42) > 10^6.

See A331940 for more information on the Hardy-Littlewood constant. The polynomials described by this sequence are increasingly prime-avoiding.

The following table provides the minimum record values of 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

       2 3.70011 10448345 3.81422

       3 2.07514  5794128 2.13869

       7 0.88360  2411224 0.91046

      13 0.87451  2344299 0.89971

      .. .......  ....... .......

  584791 0.21378   445220 0.21860

  640819 0.21229   439946 0.21641

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..41.

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, A331945, A331946, A331947.

Sequence in context: A053613 A013645 A130272 * A342529 A172238 A319496

Adjacent sequences:  A331945 A331946 A331947 * A331949 A331950 A331951

KEYWORD

nonn

AUTHOR

Hugo Pfoertner, Feb 10 2020

STATUS

approved

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Last modified May 15 17:11 EDT 2021. Contains 343920 sequences. (Running on oeis4.)