A331946 -id:A331946 - cp's OEIS frontend

A331945 Factors k > 0 such that the polynomial k*x^2 + 1 produces a record of its Hardy-Littlewood constant.

1, 2, 3, 4, 12, 18, 28, 58, 190, 462, 708, 5460, 10602, 39292, 141100, 249582, 288502

Offset: 1

Hugo Pfoertner, Feb 10 2020

a(18) > 510000.
See A331940 for more information on the Hardy-Littlewood constant. The polynomials described by this sequence have an increasing rate of generating primes.
The following table provides the record values of the Hardy-Littlewood constant C, together with the number of primes np generated by the polynomial P(x) = a(n)*x^2 + 1 for 1 <= x <= r = 10^8 and the actual ratio np*(P(r)/r)/Integral_{x=2..P(r)} 1/log(x) dx.
    a(n)    C        np    C from ratio
       1 1.37281  3954181  1.41606 (C = A199401)
       2 1.42613  4027074  1.47010
       3 1.68110  4696044  1.73337
       4 2.74563  7605407  2.82915
      12 3.36220  9037790  3.46135
      .. .......  .......  .......
  249582 7.90518 16760196  8.08633
  288502 8.21709 17367067  8.40431

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.

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]

Cf. A199401, A221712, A331940, A331941, A331946, A331948, A331948, A331949.

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

Hugo Pfoertner, Feb 10 2020

a(16) > 710000.
See A331940 for more information on the Hardy-Littlewood constant. The polynomials described by this sequence have an increasing rate of generating primes.
The following table provides the record values of the Hardy-Littlewood constant C, together with the number of primes np generated by the polynomial P(x) = a(n)*x^2 - 1 for 2 <= x <= r = 10^8 and the actual ratio np*(P(r)/r)/Integral_{x=2..P(r)} 1/log(x) dx.
    a(n)    C        np    C from ratio
       2 3.70011  10448345 3.81422
      12 4.15027  11154934 4.27219
      20 4.43326  11753085 4.56136
      68 5.01601  12883801 5.15797
      .. .......  ........ .......
  315180 7.82318  16502584 8.00057
  452042 7.85323  16434699 8.02696

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.

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]

Cf. A221712, A331940, A331941, A331945, A331946, A331948, A331949.

A331948 Nonsquare factors k > 0 such that k*x^2 - 1 produces a new minimum of its Hardy-Littlewood constant.

Original entry on oeis.org

2, 3, 7, 13, 19, 31, 79, 151, 211, 331, 499, 631, 751, 991, 1171, 2011, 2311, 2671, 3019, 3931, 4159, 4951, 5119, 6451, 7459, 10651, 18379, 32971, 48799, 61051, 78439, 84319, 162451, 199411, 230239, 257371, 404251, 462331, 529699, 584791, 640819

Offset: 1

Hugo Pfoertner, Feb 10 2020

nonn

a(42) > 10^6.
See A331940 for more information on the Hardy-Littlewood constant. The polynomials described by this sequence are increasingly prime-avoiding.
The following table provides the minimum record values of C, together with the number of primes np generated by the polynomial P(x) = a(n)*x^2 - 1 for 1 <= x <= r = 10^8 and the actual ratio np*(P(r)/r)/Integral_{x=2..P(r)} 1/log(x) dx.
    a(n)    C       np    C from ratio
       2 3.70011 10448345 3.81422
       3 2.07514  5794128 2.13869
       7 0.88360  2411224 0.91046
      13 0.87451  2344299 0.89971
      .. .......  ....... .......
  584791 0.21378   445220 0.21860
  640819 0.21229   439946 0.21641

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.

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]

Cf. A221712, A331940, A331945, A331946, A331947.

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

Hugo Pfoertner, Feb 20 2020

a(18) > 100000.
See A331940 for more information on the Hardy-Littlewood constant. The polynomials described by this sequence are increasingly avoiding primes.
The following table provides the minimum values of the Hardy-Littlewood constant C, together with the number of primes np generated by the polynomial P(x) = x^2 + a(n)*x + 1 for 1 <= x <= r = 10^8 and the actual ratio np*(P(r)/r)/Integral_{x=2..P(r)} 1/log(x) dx.
   a(n)     C       np    C from ratio
      3  3.54661 10220078 3.65998
      4  1.38342  3982973 1.42637
      8  0.91172  2627239 0.94086
     20  0.76532  2204290 0.78939
  .....  .......  ....... .......
  25220  0.39947  1151122 0.41224
  37990  0.39945  1151126 0.41224

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.

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]

Cf. A221712, A331940, A331945, A331946, A331947, A331948, A331949, A332708.

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

Hugo Pfoertner, Feb 20 2020

a(10) > 80000.
See A331940 for more information on the Hardy-Littlewood constant. The polynomials described by this sequence have an increasing rate of generating primes.
The following table provides the record values of the Hardy-Littlewood constant C, together with the number of primes np generated by the polynomial P(x) = x^2 + a(n)*x + 1 for 1 <= x <= r = 10^8 and the actual ratio np*(P(r)/r)/Integral_{x=2..P(r)} 1/log(x) dx.
  a(n)     C       np    C from ratio
     1  2.24147  6456835 2.31230
     3  3.54661 10220078 3.65998
    21  5.58679 16096923 5.76458
   231  5.74156 16543757 5.92460
   879  5.83722 16813676 6.02126
  1011  5.92725 17073610 6.11435
  1089  6.03701 17392675 6.22861
  1659  6.04359 17413761 6.23617
  2751  7.46622 21508374 7.70252

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.

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]

Cf. A221712, A331940, A331945, A331946, A331947, A331948, A331949, A332707.

A342566 Noncube factors k > 0 such that k*x^3 + 1 produces a new minimum of its Hardy-Littlewood constant.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 29, 43, 83, 239, 463, 911, 1483, 1721, 3067, 4187, 10613, 12433, 15287, 26447

Offset: 1

Hugo Pfoertner, May 03 2021

a(21)>40000.
For more information and references see A331950.
4187=53*79 is the first nonprime term.

			   n  a(n)   Hardy-Littlewood         np / (expected number of primes)
             constant (rounded)       obtained from Li(a(n)*(10^9)^3+1)
                        np (x<=10^9)  (similar to table in A331946)
   1     2   2.597079115  43503785    2.69054
   2     3   2.085815908  34707483    2.16060
   3     5   1.428347905  23566489    1.47910
   4     7   1.290763004  21179402    1.33641
   5    11   1.241279598  20211462    1.28447
  ...
  18 12433   0.450506688   6582602    0.46462
  19 15287   0.422449638   6150009    0.43536
  20 26447   0.418323708   6045844    0.43130

Cf. A331946 (similar for k*x^2+1), A331950, A342549.

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

Hugo Pfoertner, May 03 2021

a(24) > 42500.

For more information and references see A331950.

			   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

Cf. A331946, A331949 (similar for x^2+k), A331950, A342547.

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A331945 Factors k > 0 such that the polynomial k*x^2 + 1 produces a record of its Hardy-Littlewood constant.

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A331947 Factors k > 1 such that the polynomial k*x^2 - 1 produces a record of its Hardy-Littlewood constant.

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A331948 Nonsquare factors k > 0 such that k*x^2 - 1 produces a new minimum of its Hardy-Littlewood constant.

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A332707 Factors k > 2 such that the polynomial x^2 + k*x + 1 produces a new minimum of its Hardy-Littlewood constant.

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A332708 Factors k >= 0 such that the polynomial x^2 + k*x + 1 produces a record of its Hardy-Littlewood constant.

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A342566 Noncube factors k > 0 such that k*x^3 + 1 produces a new minimum of its Hardy-Littlewood constant.

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A342569 Noncube addends k > 0 such that x^3 + k produces a new minimum of its Hardy-Littlewood constant.

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Author

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Examples

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