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