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A331946 Factors k > 0 such that k*x^2 + 1 produces a new minimum of its Hardy-Littlewood constant. 5
1, 5, 11, 17, 29, 41, 89, 101, 461, 521, 761, 941, 1091, 1361, 1889, 2141, 3449, 4289, 5381, 5561, 10709, 15461, 23201, 59309, 70769, 134741, 174929, 329969, 493349 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(30) > 600000.

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 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

       5 0.66031  1816520 0.67979

      11 0.56115  1512897 0.57810

      17 0.52244  1392498 0.53816

      .. .......   ...... .......

  329969 0.20443   430342 0.20883

  493349 0.20348   424719 0.20781

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..29.

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, A331947, A331948.

Sequence in context: A067606 A184247 A046135 * A162336 A234346 A074267

Adjacent sequences:  A331943 A331944 A331945 * A331947 A331948 A331949

KEYWORD

nonn,more

AUTHOR

Hugo Pfoertner, Feb 10 2020

STATUS

approved

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Last modified April 23 01:52 EDT 2021. Contains 343198 sequences. (Running on oeis4.)