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

%I #10 Apr 30 2021 10:59:09

%S 2,3,17,74,165,205,2609,23602

%N Addends k > 0 such that the polynomial x^3 + k produces a record of its Hardy-Littlewood constant.

%C For more information and references see A331950.

%C Cubic polynomials with no quadratic terms have a poor yield in generating primes compared to quadratic polynomials. This can be seen when comparing the Hardy-Littlewood constants HL for quadratic polynomials of the form x^2 + k (k given in A003521) where HL(x^2 + 1) = 1.3728..., HL (x^2 + 7) = 1.9730..., ..., HL(x^2 + 991027) = 4.1237..., whereas the best known result for the present sequence, a(8) only leads to HL(x^3 + 23602) = 1.7167...

%e n a(n) Hardy-Littlewood

%e constant (rounded)

%e 1 2 1.298539558

%e 2 3 1.390543939

%e 3 17 1.442297580

%e 4 74 1.451456320

%e 5 165 1.589487813

%e 6 205 1.637173422

%e 7 2609 1.679828689

%e 8 23602 1.716729673

%Y Cf. A003521 (records for x^2+k), A331950.

%K nonn,hard,more

%O 1,1

%A _Hugo Pfoertner_, Apr 29 2021