Search: a332707 -id:a332707
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A332708
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Factors k >= 0 such that the polynomial x^2 + k*x + 1 produces a record of its Hardy-Littlewood constant.
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+10
1
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OFFSET
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1,2
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COMMENTS
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a(10) > 80000.
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) = x^2 + a(n)*x + 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 2.24147 6456835 2.31230
3 3.54661 10220078 3.65998
21 5.58679 16096923 5.76458
231 5.74156 16543757 5.92460
879 5.83722 16813676 6.02126
1011 5.92725 17073610 6.11435
1089 6.03701 17392675 6.22861
1659 6.04359 17413761 6.23617
2751 7.46622 21508374 7.70252
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REFERENCES
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Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
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LINKS
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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