A331945
Factors k > 0 such that the polynomial k*x^2 + 1 produces a record of its Hardy-Littlewood constant.
Original entry on oeis.org
1, 2, 3, 4, 12, 18, 28, 58, 190, 462, 708, 5460, 10602, 39292, 141100, 249582, 288502
Offset: 1
- Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
A331947
Factors k > 1 such that the polynomial k*x^2 - 1 produces a record of its Hardy-Littlewood constant.
Original entry on oeis.org
2, 12, 20, 68, 90, 98, 132, 252, 318, 362, 398, 1722, 259668, 315180, 452042
Offset: 1
- Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
A003420
Values of m in the discriminant D = -4*m leading to a new maximum of the L-function of the Dirichlet series L(1) = Sum_{k=1..oo} Kronecker(D,k)/k.
Original entry on oeis.org
1, 2, 5, 11, 14, 26, 41, 89, 101, 194, 314, 341, 689, 1091, 1154, 1889, 2141, 3449, 3506, 5561, 6254, 8126, 8774, 10709, 13166, 15461, 23201, 24569, 30014, 81149, 81626, 162686, 243374, 644474, 839354, 879941
Offset: 1
a(1) = 1: L(1) for D=-4*1 ~= 0.785398... = Pi/4.
a(2) = 2: L(1) for D=-4*2 ~= 1.11072073... = Pi/(2*sqrt(2)), a(2) > a(1);
L(1) for D=-4*3 ~= 0.90689..., L(1) for D=-4*4 ~= 0.785398..., both < a(2);
a(3) = 5: L(1) for D=-4*5 = 1.40496..., a(3) > a(2).
- D. Shanks, Systematic examination of Littlewood's bounds on L(1,chi), pp. 267-283 of Analytic Number Theory, ed. H. G. Diamond, Proc. Sympos. Pure Math., 24 (1973). Amer. Math. Soc.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Cf.
A331949, which has almost identical terms.
New title, a(1) prepended, missing term 23201 and a(29)-a(33) from
Hugo Pfoertner, Feb 02 2020
A332707
Factors k > 2 such that the polynomial x^2 + k*x + 1 produces a new minimum of its Hardy-Littlewood constant.
Original entry on oeis.org
3, 4, 8, 20, 40, 230, 260, 680, 1910, 2120, 6670, 9710, 10310, 23500, 25220, 37990, 71800
Offset: 1
- Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
A332708
Factors k >= 0 such that the polynomial x^2 + k*x + 1 produces a record of its Hardy-Littlewood constant.
Original entry on oeis.org
1, 3, 21, 231, 879, 1011, 1089, 1659, 2751
Offset: 1
- Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
A342569
Noncube addends k > 0 such that x^3 + k produces a new minimum of its Hardy-Littlewood constant.
Original entry on oeis.org
2, 5, 6, 13, 15, 20, 34, 83, 174, 246, 911, 1065, 1084, 1455, 1490, 1546, 3674, 8644, 9556, 15287, 15378, 15826, 25670
Offset: 1
n a(n) Hardy-Littlewood np / (expected number of primes)
constant (rounded) obtained from Li((10^9)^3+a(n))
np (x<=10^9) (similar to table in A331946)
1 2 1.298539558 22009948 1.34597
2 5 1.142678324 19372839 1.18470
3 6 0.822719287 13944026 0.85272
4 13 0.814418714 13802244 0.84405
5 15 0.784789179 13305075 0.81364
...
20 15287 0.422422003 7162493 0.43801
21 15378 0.419380705 7108723 0.43472
22 15826 0.416982640 7068923 0.43228
23 25670 0.388993112 6597073 0.40343
Showing 1-6 of 6 results.
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