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 A331949 Addends k > 0 such that x^2 + k produces a new minimum of its Hardy-Littlewood Constant. 5
 1, 2, 5, 11, 14, 26, 41, 89, 101, 194, 314, 341, 446, 689, 1091, 1154, 1889, 2141, 3449, 3506, 5561, 6254, 8126, 8774, 10709, 13166, 15461, 23201, 24569, 30014, 81626, 162686 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS This sequence is almost identical to A003420. However, there is an additional term 426 and after 30014 the number 81626 follows, while in A003420, 81149 is present between 30014 and 81626. With C(m) = Product_{p=primes} 1 - Kronecker(-4*m,p)/(p - 1) (Hardy-Littlewood) L1(m) = Sum_{j>0} Kronecker(-4*m,j)/j (L-function of the Dirichlet series) the following table shows the differences:               Criterion           decrease increase       k      C        L1      341  0.28309  2.38177      446  0.28272  2.38014 not in A003420 because L1(446) < L1(341)      689  0.28193  2.39370    ...    30014  0.21541  3.08274    81149  0.21560  3.08792 not in this sequence because C(81149) > C(30014)    81626  0.20883  3.17785   162686  0.20478  3.24017 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. Henri Cohen, High-precision computation of Hardy-Littlewood constants, preprint, 1998. [pdf copy, with permission] D. Shanks, Systematic examination of Littlewood's bounds on L(1,chi), Proc. Sympos. Pure Math., 24 (1973). Amer. Math. Soc. (Annotated scanned copy) PROG (PARI) \\ The function HardyLittlewood2 is provided at the Belabas, Cohen link. hl2min=0; for(add=1, 500, my(hl=HardyLittlewood2(n^2+add)); if(hl

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Last modified May 14 12:37 EDT 2021. Contains 343884 sequences. (Running on oeis4.)