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A331940 Addends k > 0 such that the polynomial x^2 + x + k produces a record of its Hardy-Littlewood Constant.

%I A331940 #23 Mar 19 2025 05:54:30
%S A331940 1,11,17,41,21377,27941,41537,55661,115721,239621,247757
%N A331940 Addends k > 0 such that the polynomial x^2 + x + k produces a record of its Hardy-Littlewood Constant.
%C A331940 The Hardy and Littlewood Conjecture F provides an estimate of the number of primes generated by a quadratic polynomial P(x) for 0 <= x <= m in the form C * Integral_{x=2..m} 1/log(x) dx, with C given by an Euler product that is a function of the fundamental discriminant of the polynomial. Cohen describes an efficient method for the computation of C.
%C A331940 The following table provides the record values of C, together with the number of primes np generated by the polynomial x^2 + x + a(n) for x <= 10^8 and the actual ratio 2*np/Integral_{x=2..10^8} 1/log(x) dx.
%C A331940     a(n)    C       np    C from ratio
%C A331940        1 2.24147  6456835 2.24110
%C A331940       11 3.25944  9389795 3.25910
%C A331940       17 4.17466 12027453 4.17460
%C A331940       41 6.63955 19132653 6.64073
%C A331940    21377 6.92868 19962992 6.92894
%C A331940    27941 7.26400 20931145 7.26497
%C A331940    41537 7.32220 21092134 7.32085
%C A331940    55661 7.45791 21483365 7.45664
%C A331940   115721 7.70935 22210771 7.70912
%C A331940   239621 7.72932 22268336 7.72909
%C A331940   247757 8.24741 23762118 8.24757
%C A331940 Jacobson and Williams found significantly larger values of C for very large addends k, e.g. C = 2*5.36708 = 10.73416 for k = 3399714628553118047.
%D A331940 Keith Conrad, Hardy-Littlewood Constants. In: Mathematical Properties of Sequences and Other Combinatorial Structures, eds. Jong-Seon No, Hong-Yeop Song, Tor Helleseth, P. Vijay Kumar, Springer New York, 2003, pages 133-154.
%H A331940 Karim Belabas and Henri Cohen, <a href="/A221712/a221712.gp.txt">Computation of the Hardy-Littlewood constant for quadratic polynomials</a>, PARI/GP script, 2020.
%H A331940 Henri Cohen, <a href="/A221712/a221712.pdf">High precision computation of Hardy-Littlewood constants</a>. [Cached pdf version, with permission]
%H A331940 Keith Conrad, <a href="https://kconrad.math.uconn.edu/articles/hlconst.pdf">Hardy-Littlewood Constants</a>, (2003).
%H A331940 Michael J. Jacobson Jr. and Hugh C. Williams, <a href="https://doi.org/10.1090/S0025-5718-02-01418-7">New Quadratic Polynomials With High Densities Of Prime Values</a>, Math. Comp., 72, 241, 499-519, 2002.
%o A331940 (PARI) \\ The function HardyLittlewood2 is provided at the Belabas, Cohen link.
%o A331940 hl2max=0; for(add=0,100,my(hl=HardyLittlewood2(n^2+n+add)); if(hl>hl2max,print1(add,", "); hl2max=hl))
%Y A331940 Cf. A002837, A007635, A014556, A116206, A331877.
%Y A331940 Cf. A221712, A221713 (Constants C including factor 1/2).
%K A331940 nonn,more,hard
%O A331940 1,2
%A A331940 _Hugo Pfoertner_, Feb 02 2020