A331947 -id:A331947 - cp's OEIS frontend

A331946 Factors k > 0 such that k*x^2 + 1 produces a new minimum of its Hardy-Littlewood constant.

1, 5, 11, 17, 29, 41, 89, 101, 461, 521, 761, 941, 1091, 1361, 1889, 2141, 3449, 4289, 5381, 5561, 10709, 15461, 23201, 59309, 70769, 134741, 174929, 329969, 493349

Offset: 1

Hugo Pfoertner, Feb 10 2020

a(30) > 600000.
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 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
       5 0.66031  1816520 0.67979
      11 0.56115  1512897 0.57810
      17 0.52244  1392498 0.53816
      .. .......   ...... .......
  329969 0.20443   430342 0.20883
  493349 0.20348   424719 0.20781

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, A331947, A331948.

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.

A338477 Numbers k such that 398*k^2 - 1 is prime.

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 17, 18, 19, 20, 22, 24, 25, 27, 28, 29, 33, 34, 37, 38, 43, 44, 46, 47, 51, 52, 54, 55, 58, 59, 60, 67, 68, 71, 73, 75, 79, 80, 81, 82, 83, 85, 86, 87, 89, 90, 93, 94, 95, 96, 97, 100, 103, 106, 107, 108, 110, 112, 114, 116, 117, 119, 121, 124, 125, 128

Offset: 1

Robert Israel, Oct 29 2020

nonn

There are 414 such primes for 1 <= x <= 1000, and 3280 for 1 <= x <= 10000.

			a(3)=4 is in the sequence because 398*4^2 - 1 = 6367 is prime.

Robert Israel, Table of n, a(n) for n = 1..10000
V. Granville, Quadratic progressions with very high prime density, MathOverflow.

Cf. A338476.

Cf. A331947, where 398 is a term.

select(t -> isprime(398*t^2-1), [$1..1000]);

a(n) = sqrt(A338476(n) + 1)/398.

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.

A338476 Primes of the form 398*x^2-1.

Original entry on oeis.org

397, 3581, 6367, 9949, 14327, 19501, 25471, 32237, 39799, 48157, 67261, 78007, 115021, 128951, 143677, 159199, 192631, 229247, 248749, 290141, 312031, 334717, 433421, 460087, 544861, 574711, 735901, 770527, 842167, 879181, 1035197, 1076191, 1160567, 1203949, 1338871, 1385437, 1432799, 1786621

Offset: 1

Robert Israel, Oct 29 2020

nonn

There are 414 such primes for 1 <= x <= 1000, and 3280 for 1 <= x <= 10000.

			a(3) = 398*4^2-1 = 6367 is prime.

V. Granville, Quadratic progressions with very high prime density, MathOverflow.

Cf. A331947(11)=398, A338477.

select(isprime, [seq(398*x^2-1,x=1..1000)]);

Select[398 Range[100]^2-1,PrimeQ] (* Harvey P. Dale, Jan 13 2023 *)

a(n) = 398*A338477(n)^2-1.

cp's OEIS Frontend

A331946 Factors k > 0 such that k*x^2 + 1 produces a new minimum of its Hardy-Littlewood constant.

Views

Author

Keywords

Comments

References

Links

Crossrefs

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

A338477 Numbers k such that 398*k^2 - 1 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

A338476 Primes of the form 398*x^2-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula