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A332707 Factors k > 2 such that the polynomial x^2 + k*x + 1 produces a new minimum of its Hardy-Littlewood constant. 1
3, 4, 8, 20, 40, 230, 260, 680, 1910, 2120, 6670, 9710, 10310, 23500, 25220, 37990, 71800 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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
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
Sequence in context: A254715 A107328 A065034 * A129285 A051440 A101932
KEYWORD
nonn,more
AUTHOR
Hugo Pfoertner, Feb 20 2020
STATUS
approved

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Last modified April 23 06:45 EDT 2024. Contains 371906 sequences. (Running on oeis4.)