A331950 -id:A331950 - cp's OEIS frontend

A050266 Primes of the form n^3 + n^2 + 17, for nonnegative values of n.

17, 19, 29, 53, 97, 167, 269, 409, 593, 827, 1117, 1889, 2383, 2957, 3617, 6173, 7237, 9719, 11149, 12713, 16267, 18269, 25247, 27917, 33809, 47969, 56333, 65617, 70619, 75869, 81373, 99469, 112913, 120067, 143329, 151703, 160397, 188459

Offset: 1

Eric W. Weisstein

n=-2 produces the prime 13. - Robert Price, Apr 25 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.

Cf. A331950.

[a: n in [0..100] | IsPrime(a) where a is  n^3+n^2+17]; // Vincenzo Librandi, Dec 08 2011

f:= seq(n^3+n^2+17, n = 0..100); select(isprime, f); # G. C. Greubel, Feb 06 2020

Select[Table[n^3+n^2+17,{n,0,100}],PrimeQ] (* Vincenzo Librandi, Dec 08 2011 *)

for(n=0, 100, my(m=n^3+n^2+17); if(isprime(m), print1(m ", "))) \\ G. C. Greubel, Feb 06 2020

[n^3+n^2+17 for n in (0..100) if is_prime(n^3+n^2+17) ] # G. C. Greubel, Feb 06 2020

17 added by Vincenzo Librandi, Dec 08 2011

Altered definition to require that n be nonnegative. - Robert Price, Apr 25 2016

A342547 Addends k > 0 such that the polynomial x^3 + k produces a record of its Hardy-Littlewood constant.

Original entry on oeis.org

2, 3, 17, 74, 165, 205, 2609, 23602

Offset: 1

Hugo Pfoertner, Apr 29 2021

For more information and references see A331950.

Cubic polynomials with no quadratic terms have a poor yield in generating primes compared to quadratic polynomials. This can be seen when comparing the Hardy-Littlewood constants HL for quadratic polynomials of the form x^2 + k (k given in A003521) where HL(x^2 + 1) = 1.3728..., HL (x^2 + 7) = 1.9730..., ..., HL(x^2 + 991027) = 4.1237..., whereas the best known result for the present sequence, a(8) only leads to HL(x^3 + 23602) = 1.7167...

			  n  a(n)   Hardy-Littlewood
            constant (rounded)
  1     2   1.298539558
  2     3   1.390543939
  3    17   1.442297580
  4    74   1.451456320
  5   165   1.589487813
  6   205   1.637173422
  7  2609   1.679828689
  8 23602   1.716729673

Cf. A003521 (records for x^2+k), A331950.

A342549 Factors k > 0 such that the polynomial 2kx^3 + 1 produces a record of its Hardy-Littlewood constant.

Original entry on oeis.org

1, 5, 12, 57, 75, 135, 195, 210, 660

Offset: 1

Hugo Pfoertner, May 01 2021

a(10) > 10000.

For more information and references see A331950.

			   n  a(n)   Hardy-Littlewood
             constant (rounded)
   1     1   2.5970791151
   2     5   3.3401203819
   3    12   4.1716318164
   4    57   4.5504531392
   5    75   4.7090460169
   6   135   5.0101805728
   7   195   5.3408627801
   8   210   6.0848257259
   9   660   6.5566372288

Cf. A331945 (records for k*x^2+1), A331950, A342547, A342548, A342566.

A342548 Factors k > 0 such that the polynomial k*x^3 + x^2 + 1 produces a record of its Hardy-Littlewood constant.

Original entry on oeis.org

1, 3, 21, 69, 81, 99, 489, 8169

Offset: 1

Hugo Pfoertner, May 01 2021

For more information and references see A331950.

			  n  a(n)   Hardy-Littlewood
            constant (rounded)
  1     1   3.0750320679
  2     3   3.3225199464
  3    21   4.3206296013
  4    69   4.3659028300
  5    81   4.9042526822
  6    99   6.0219444490
  7   489   6.5115238241
  8  8169   6.7898774153

Cf. A331950, A342547, A342549.

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.

cp's OEIS Frontend

A050266 Primes of the form n^3 + n^2 + 17, for nonnegative values of n.

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A342547 Addends k > 0 such that the polynomial x^3 + k produces a record of its Hardy-Littlewood constant.

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A342549 Factors k > 0 such that the polynomial 2*k*x^3 + 1 produces a record of its Hardy-Littlewood constant.

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A342548 Factors k > 0 such that the polynomial k*x^3 + x^2 + 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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A342549 Factors k > 0 such that the polynomial 2kx^3 + 1 produces a record of its Hardy-Littlewood constant.