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A331949 Addends k > 0 such that x^2 + k produces a new minimum of its Hardy-Littlewood Constant.

%I A331949 #9 Aug 26 2022 18:50:07
%S A331949 1,2,5,11,14,26,41,89,101,194,314,341,446,689,1091,1154,1889,2141,
%T A331949 3449,3506,5561,6254,8126,8774,10709,13166,15461,23201,24569,30014,
%U A331949 81626,162686
%N A331949 Addends k > 0 such that x^2 + k produces a new minimum of its Hardy-Littlewood Constant.
%C A331949 This sequence is almost identical to A003420. However, there is an additional term 446 and after 30014 the number 81626 follows, while in A003420, 81149 is present between 30014 and 81626. With
%C A331949 C(m) = Product_{p=primes} 1 - Kronecker(-4*m,p)/(p - 1) (Hardy-Littlewood)
%C A331949 L1(m) = Sum_{j>0} Kronecker(-4*m,j)/j (L-function of the Dirichlet series)
%C A331949 the following table shows the differences:
%C A331949               Criterion
%C A331949           decrease increase
%C A331949       k      C        L1
%C A331949      341  0.28309  2.38177
%C A331949      446  0.28272  2.38014 not in A003420 because L1(446) < L1(341)
%C A331949      689  0.28193  2.39370
%C A331949    ...
%C A331949    30014  0.21541  3.08274
%C A331949    81149  0.21560  3.08792 not in this sequence because C(81149) > C(30014)
%C A331949    81626  0.20883  3.17785
%C A331949   162686  0.20478  3.24017
%D A331949 Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
%H A331949 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 A331949 Henri Cohen, <a href="/A221712/a221712.pdf">High-precision computation of Hardy-Littlewood constants</a>, preprint, 1998. [pdf copy, with permission]
%H A331949 D. Shanks, <a href="/A003419/a003419.pdf">Systematic examination of Littlewood's bounds on L(1,chi)</a>, Proc. Sympos. Pure Math., 24 (1973). Amer. Math. Soc. (Annotated scanned copy)
%o A331949 (PARI) \\ The function HardyLittlewood2 is provided at the Belabas, Cohen link.
%o A331949 hl2min=oo; for(add=1,500,my(hl=HardyLittlewood2(n^2+add));if(hl<hl2min,print1(add,", ");hl2min=hl))
%Y A331949 Cf. A003420, A003521, A331940, A331941.
%K A331949 nonn,more
%O A331949 1,2
%A A331949 _Hugo Pfoertner_, Feb 04 2020