A331950 Addends k > 0 such that the polynomial x^3 + x^2 + k produces a record of its Hardy-Littlewood Constant.

1, 17, 101, 1487, 13301, 19421, 91127

Offset: 1

Author

Hugo Pfoertner, Feb 04 2020

Keywords

• nonn
• more
• hard

Comments

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.

k C np C(k)/C(1) np(k)/np(1)

1 3.075032 5907486 1.0000000 1.0000000

17 5.653199 10860984 1.8384196 1.8385120

101 6.035464 11594890 1.9627322 1.9627452

1487 6.783304 13030949 2.2059297 2.2058366

13301 6.890698 13236230 2.2408541 2.2405859

19421 6.967707 13380959 2.2658974 2.2650852

91127 7.121020 13682111 2.3157547 2.3160632

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.
• Karim Belabas, Henri Cohen, Computation of the Hardy-Littlewood constant for cubic polynomials, PARI/GP script, 2020.
• Henri Cohen, High precision computation of Hardy-Littlewood constants, preprint, 1998. [pdf copy, with permission]
• Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.
• Wikipedia, Bateman-Horn conjecture.

Crossrefs

Cf. A050266, A221712, A331940.

Programs

PARI

\\ The functions HardyLittlewood2 and HardyLittlewood3 are provided at the Belabas, Cohen links.
hl3max=0; for(add=0,101,my(hl=HardyLittlewood3(n^3+n^2+add));if(hl>hl3max,print1(add,", ");hl3max=hl))