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A331946
Factors k > 0 such that k*x^2 + 1 produces a new minimum of its Hardy-Littlewood constant.
7
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
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
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
KEYWORD
nonn,more
AUTHOR
Hugo Pfoertner, Feb 10 2020
STATUS
approved