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A331948
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Nonsquare factors k > 0 such that k*x^2 - 1 produces a new minimum of its Hardy-Littlewood constant.
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5
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2, 3, 7, 13, 19, 31, 79, 151, 211, 331, 499, 631, 751, 991, 1171, 2011, 2311, 2671, 3019, 3931, 4159, 4951, 5119, 6451, 7459, 10651, 18379, 32971, 48799, 61051, 78439, 84319, 162451, 199411, 230239, 257371, 404251, 462331, 529699, 584791, 640819
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OFFSET
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1,1
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COMMENTS
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a(42) > 10^6.
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 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
2 3.70011 10448345 3.81422
3 2.07514 5794128 2.13869
7 0.88360 2411224 0.91046
13 0.87451 2344299 0.89971
.. ....... ....... .......
584791 0.21378 445220 0.21860
640819 0.21229 439946 0.21641
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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
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AUTHOR
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STATUS
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approved
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