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A332707
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Factors k > 2 such that the polynomial x^2 + k*x + 1 produces a new minimum of its Hardy-Littlewood constant.
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1
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3, 4, 8, 20, 40, 230, 260, 680, 1910, 2120, 6670, 9710, 10310, 23500, 25220, 37990, 71800
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OFFSET
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1,1
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COMMENTS
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a(18) > 100000.
See A331940 for more information on the Hardy-Littlewood constant. The polynomials described by this sequence are increasingly avoiding primes.
The following table provides the minimum 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
3 3.54661 10220078 3.65998
4 1.38342 3982973 1.42637
8 0.91172 2627239 0.94086
20 0.76532 2204290 0.78939
..... ....... ....... .......
25220 0.39947 1151122 0.41224
37990 0.39945 1151126 0.41224
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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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