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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
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]
KEYWORD
nonn,more,hard
AUTHOR
Hugo Pfoertner, Feb 10 2020
STATUS
approved