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A331947
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Factors k > 1 such that the polynomial k*x^2 - 1 produces a record of its Hardy-Littlewood constant.
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6
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2, 12, 20, 68, 90, 98, 132, 252, 318, 362, 398, 1722, 259668, 315180, 452042
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OFFSET
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1,1
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COMMENTS
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a(16) > 710000.
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 2 <= 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
12 4.15027 11154934 4.27219
20 4.43326 11753085 4.56136
68 5.01601 12883801 5.15797
.. ....... ........ .......
315180 7.82318 16502584 8.00057
452042 7.85323 16434699 8.02696
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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,hard
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AUTHOR
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STATUS
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approved
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