A332708 -id:A332708 - cp's OEIS frontend

A332707 Factors k > 2 such that the polynomial x^2 + k*x + 1 produces a new minimum of its Hardy-Littlewood constant.

3, 4, 8, 20, 40, 230, 260, 680, 1910, 2120, 6670, 9710, 10310, 23500, 25220, 37990, 71800

Offset: 1

Hugo Pfoertner, Feb 20 2020

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

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.

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]

Cf. A221712, A331940, A331945, A331946, A331947, A331948, A331949, A332708.

cp's OEIS Frontend

A332707 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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