A331940 -id:A331940 - 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.

A331950 Addends k > 0 such that the polynomial x^3 + x^2 + k produces a record of its Hardy-Littlewood Constant.

Original entry on oeis.org

1, 17, 101, 1487, 13301, 19421, 91127

Offset: 1

Hugo Pfoertner, Feb 04 2020

The method for calculating the Hardy-Littlewood constant for quadratic polynomials can be generalized to cubic polynomials (see the preprint by H. Cohen for the exact definition). In this case too, the constant is an estimate of which fraction (e.g. in relation to a random placement) of prime numbers the polynomial hits within its range of values. The following table shows that the ratio of the actual prime number hits for 1 <= x <= 10^8 for different addend values corresponds almost exactly to the ratio of the Hardy-Littlewood constants. The Hardy-Littlewood constant C and the number of prime hits np at offset = 1 are chosen as reference values.
        k     C       np     C(k)/C(1)  np(k)/np(1)
        1 3.075032  5907486  1.0000000  1.0000000
       17 5.653199 10860984  1.8384196  1.8385120
      101 6.035464 11594890  1.9627322  1.9627452
     1487 6.783304 13030949  2.2059297  2.2058366
    13301 6.890698 13236230  2.2408541  2.2405859
    19421 6.967707 13380959  2.2658974  2.2650852
    91127 7.121020 13682111  2.3157547  2.3160632

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.
Karim Belabas, Henri Cohen, Computation of the Hardy-Littlewood constant for cubic polynomials, PARI/GP script, 2020.
Henri Cohen, High precision computation of Hardy-Littlewood constants, preprint, 1998. [pdf copy, with permission]
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.
Wikipedia, Bateman-Horn conjecture.

Cf. A050266, A221712, A331940.

\\ The functions HardyLittlewood2 and HardyLittlewood3 are provided at the Belabas, Cohen links.
hl3max=0; for(add=0,101,my(hl=HardyLittlewood3(n^3+n^2+add));if(hl>hl3max,print1(add,", ");hl3max=hl))

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

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.

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

Original entry on oeis.org

1, 2, 5, 11, 14, 26, 41, 89, 101, 194, 314, 341, 446, 689, 1091, 1154, 1889, 2141, 3449, 3506, 5561, 6254, 8126, 8774, 10709, 13166, 15461, 23201, 24569, 30014, 81626, 162686

Offset: 1

Hugo Pfoertner, Feb 04 2020

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(m) = Product_{p=primes} 1 - Kronecker(-4*m,p)/(p - 1) (Hardy-Littlewood)
L1(m) = Sum_{j>0} Kronecker(-4*m,j)/j (L-function of the Dirichlet series)
the following table shows the differences:
              Criterion
          decrease increase
      k      C        L1
     341  0.28309  2.38177
     446  0.28272  2.38014 not in A003420 because L1(446) < L1(341)
     689  0.28193  2.39370
   ...
   30014  0.21541  3.08274
   81149  0.21560  3.08792 not in this sequence because C(81149) > C(30014)
   81626  0.20883  3.17785
  162686  0.20478  3.24017

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]
D. Shanks, Systematic examination of Littlewood's bounds on L(1,chi), Proc. Sympos. Pure Math., 24 (1973). Amer. Math. Soc. (Annotated scanned copy)

Cf. A003420, A003521, A331940, A331941.

\\ The function HardyLittlewood2 is provided at the Belabas, Cohen link.
hl2min=oo; for(add=1,500,my(hl=HardyLittlewood2(n^2+add));if(hl

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.

A356751 Positive integers m such that x^2 - x + m contains more than m/2 prime numbers for x = 1, 2, ..., m.

Original entry on oeis.org

3, 5, 7, 11, 17, 41, 47, 59, 67, 101, 107, 161, 221, 227, 347, 377

Offset: 1

Marco Ripà, Aug 25 2022

This sequence is related to A188424, since we are considering only the addends m := 2n - 1 of k^2 + k + 2n - 1 such that A188424(n)/(2n - 1) > 1/2.
Furthermore, it is conjectured that the present sequence consists of only 16 terms (it has been checked by brute force that there are only 16 terms which are smaller than 20000) and that they are all prime or semiprime (e.g., a(12) = 161, a(13) = 221, and a(16) = 377 are semiprime). Lastly, we trivially point out that all terms must be odd, since if m is even, then x^2 - x + m is also even (and x^2 - x + 2 has only one prime for x <= 2).
For an explanation of the abundance of primes of the form x^2 - x + m, for some given m, see Goudsmit's paper in Links.
Stronger conjecture: for every real number e > 0 and every integer m > 0, there are finitely many integer polynomials P(x) = Ax^2 + Bx + C with at least e*m primes in P(1), ..., P(m) and max(|A|, |B|, |C|) <= m. - Charles R Greathouse IV, Sep 11 2022
Altering the bounds for x in the definition to 0 <= x <= m-1 (and counting the same prime twice for x=0 and x=1 if m is prime) would result in an additional term 2. Conjecturally, there would be no more additional terms. - Pontus von Brömssen, Jun 20 2024

			7 is a term since x^2 - x + 7 is prime for x = 1, 3, 4, and 6, which is 4 values of x, and 4 > 7/2.

S. A. Goudsmit, Unusual Prime Number Sequences, Nature Vol. 214 (1967), 1164.
Brady Haran and Matt Parker, Caboose Numbers, Youtube video, June 2024.

Cf. A005846, A007641, A007635, A057604, A188424, A331940, A356756.

Cf. A014556 (Euler's "Lucky" numbers).

q[k_] := Count[Range[k], ?(PrimeQ[#^2 - # + k] &)] > k/2; Select[Range[400], q] (* _Amiram Eldar, Aug 26 2022 *)

isok(m) = sum(k=1, m, isprime(k^2 - k + m)) > m/2; \\ Michel Marcus, Aug 26 2022

from sympy import isprime
def ok(m): return 2*sum(1 for x in range(1, m+1) if isprime(x**2-x+m)) > m
print([m for m in range(1, 400) if ok(m)]) # Michael S. Branicky, Aug 26 2022

A356756 Positive integers m such that x^2 + x + m contains at least m/2 prime numbers for x = 1, 2, ..., m.

Original entry on oeis.org

1, 5, 11, 17, 41, 47, 59, 67, 101, 107, 161, 221, 227, 347, 377

Offset: 1

Marco Ripà, Aug 26 2022

This sequence is related to A188424, since we are considering only the addends m := 2n - 1 of k^2 + k + 2n - 1 such that A188424(n)/(2n - 1) > 1/2.
It is not a subsequence of A356751, nor vice versa, since 1 is a peculiar term, whereas 3 and 7 do not belong to the present sequence, even if they are terms of A356751.
Furthermore, it is conjectured that the present sequence consists of only 15 terms (it has been checked by brute force that there are only 15 terms which are smaller than 20000). Lastly, we trivially point out that all terms must be odd, since if m is even, then x^2 + x + m is also even.
We trivially note that all the terms are odd (since x^2 + x + 2 is not prime for x = 1, nor for x = 2) and a(n - 1) = A356751(n) holds for every n > 3.
For an explanation of the abundance of primes of the form x^2 + x + m, for some given m, see Goudsmit's paper in Links.

			17 is a term since x^2 + x + 17 is prime for x = 1, 2, 3, ..., 15, which is 15 values of x, and 15 >= 17/2.

S. A. Goudsmit, Unusual Prime Number Sequences, Nature Vol. 214 (1967), 1164.

Cf. A005846, A007641, A007635, A057604, A188424, A331940, A356751.

q[k_] := Count[Range[k], ?(PrimeQ[#^2 + # + k] &)] >= k/2; Select[Range[400], q] (* _Amiram Eldar, Aug 26 2022 *)

isok(m) = sum(k=1, m, isprime(k^2 + k + m)) > m/2; \\ Michel Marcus, Aug 26 2022

from sympy import isprime
def ok(m): return 2*sum(1 for x in range(1, m+1) if isprime(x**2+x+m)) >= m
print([m for m in range(1, 400) if ok(m)]) # Michael S. Branicky, Aug 26 2022

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

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A331950 Addends k > 0 such that the polynomial x^3 + x^2 + k produces a record of its Hardy-Littlewood Constant.

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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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A331949 Addends k > 0 such that x^2 + k produces a new minimum 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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A356751 Positive integers m such that x^2 - x + m contains more than m/2 prime numbers for x = 1, 2, ..., m.

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