This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A331950 #13 Feb 16 2025 08:33:59 %S A331950 1,17,101,1487,13301,19421,91127 %N A331950 Addends k > 0 such that the polynomial x^3 + x^2 + k produces a record of its Hardy-Littlewood Constant. %C A331950 The method for calculating the Hardy-Littlewood constant for quadratic polynomials can be generalized to cubic polynomials (see the preprint by H. Cohen for the exact definition). In this case too, the constant is an estimate of which fraction (e.g. in relation to a random placement) of prime numbers the polynomial hits within its range of values. The following table shows that the ratio of the actual prime number hits for 1 <= x <= 10^8 for different addend values corresponds almost exactly to the ratio of the Hardy-Littlewood constants. The Hardy-Littlewood constant C and the number of prime hits np at offset = 1 are chosen as reference values. %C A331950 k C np C(k)/C(1) np(k)/np(1) %C A331950 1 3.075032 5907486 1.0000000 1.0000000 %C A331950 17 5.653199 10860984 1.8384196 1.8385120 %C A331950 101 6.035464 11594890 1.9627322 1.9627452 %C A331950 1487 6.783304 13030949 2.2059297 2.2058366 %C A331950 13301 6.890698 13236230 2.2408541 2.2405859 %C A331950 19421 6.967707 13380959 2.2658974 2.2650852 %C A331950 91127 7.121020 13682111 2.3157547 2.3160632 %D A331950 Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209. %H A331950 Karim Belabas, Henri Cohen, <a href="/A221712/a221712.gp.txt">Computation of the Hardy-Littlewood constant for quadratic polynomials</a>, PARI/GP script, 2020. %H A331950 Karim Belabas, Henri Cohen, <a href="/A331950/a331950.gp.txt">Computation of the Hardy-Littlewood constant for cubic polynomials</a>, PARI/GP script, 2020. %H A331950 Henri Cohen, <a href="/A221712/a221712.pdf">High precision computation of Hardy-Littlewood constants</a>, preprint, 1998. [pdf copy, with permission] %H A331950 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-Generating Polynomial.</a> %H A331950 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bateman%E2%80%93Horn_conjecture">Bateman-Horn conjecture</a>. %o A331950 (PARI) \\ The functions HardyLittlewood2 and HardyLittlewood3 are provided at the Belabas, Cohen links. %o A331950 hl3max=0; for(add=0,101,my(hl=HardyLittlewood3(n^3+n^2+add));if(hl>hl3max,print1(add,", ");hl3max=hl)) %Y A331950 Cf. A050266, A221712, A331940. %K A331950 nonn,more,hard %O A331950 1,2 %A A331950 _Hugo Pfoertner_, Feb 04 2020