A116206 -id:A116206 - cp's OEIS frontend

A050267 Primes or negative values of primes in the sequence b(n) = 47n^2 - 1701n + 10181, n >= 0.

10181, 8527, 6967, 5501, 4129, 2851, 1667, 577, -419, -1321, -2129, -2843, -3463, -3989, -4421, -4759, -5003, -5153, -5209, -5171, -5039, -4813, -4493, -4079, -3571, -2969, -2273, -1483, -599, 379, 1451, 2617, 3877, 5231, 6679, 8221, 9857, 11587, 13411, 15329, 17341, 19447, 21647, 31387

Offset: 1

Eric W. Weisstein

Terms are listed in the order of their appearance in sequence b.

This is a transformed version of the polynomial P(x) = 47*x^2 + 9*x - 5209 whose absolute value gives 43 distinct primes for -24 <= x <= 18, found by G. W. Fung in 1988. - Hugo Pfoertner, Dec 13 2019

R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004 (ISBN 0-387-20860-7); see Section A17, p. 59.
Paulo Ribenboim, The Little Book of Bigger Primes, Second Edition, Springer-Verlag New York, 2004. See p. 147.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
G. W. Fung and H. C. Williams, Quadratic polynomials which have a high density of prime values, Math. Comput. 55(191) (1990), 345-353.
Carlos Rivera, Problem 12: Prime producing polynomials, The Prime Puzzles & Problems Connection.
Jitender Singh, Prime numbers and factorization of polynomials, arXiv:2411.18366 [math.NT], 2024.
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.

Cf. A002383, A005471, A005846, A007635, A022464, A027753, A027755, A027758, A048059, A050267, A050268, A116206, A117081, A267252.

lst={};Do[p=47*n^2-1701*n+10181;If[PrimeQ[p],AppendTo[lst,p]],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 29 2009 *)
Select[Table[47n^2-1701n+10181,{n,0,50}],PrimeQ] (* Harvey P. Dale, Oct 03 2011 *)

[n | n <- apply(m->47*m^2-1701*m+10181, [0..100]), isprime(abs(n))] \\ Charles R Greathouse IV, Jun 18 2017

Edited by N. J. A. Sloane, May 10 2007
Further edited by Klaus Brockhaus, Mar 20 2010
More terms (to distinguish from quadratic) from Charles R Greathouse IV, Jun 18 2017

A331940 Addends k > 0 such that the polynomial x^2 + x + k produces a record of its Hardy-Littlewood Constant.

Original entry on oeis.org

1, 11, 17, 41, 21377, 27941, 41537, 55661, 115721, 239621, 247757

Offset: 1

Hugo Pfoertner, Feb 02 2020

The Hardy and Littlewood Conjecture F provides an estimate of the number of primes generated by a quadratic polynomial P(x) for 0 <= x <= m in the form C * Integral_{x=2..m} 1/log(x) dx, with C given by an Euler product that is a function of the fundamental discriminant of the polynomial. Cohen describes an efficient method for the computation of C.
The following table provides the record values of C, together with the number of primes np generated by the polynomial x^2 + x + a(n) for x <= 10^8 and the actual ratio 2*np/Integral_{x=2..10^8} 1/log(x) dx.
    a(n)    C       np    C from ratio
       1 2.24147  6456835 2.24110
      11 3.25944  9389795 3.25910
      17 4.17466 12027453 4.17460
      41 6.63955 19132653 6.64073
   21377 6.92868 19962992 6.92894
   27941 7.26400 20931145 7.26497
   41537 7.32220 21092134 7.32085
   55661 7.45791 21483365 7.45664
  115721 7.70935 22210771 7.70912
  239621 7.72932 22268336 7.72909
  247757 8.24741 23762118 8.24757
Jacobson and Williams found significantly larger values of C for very large addends k, e.g. C = 2*5.36708 = 10.73416 for k = 3399714628553118047.

Keith Conrad, Hardy-Littlewood Constants. In: Mathematical Properties of Sequences and Other Combinatorial Structures, eds. Jong-Seon No, Hong-Yeop Song, Tor Helleseth, P. Vijay Kumar, Springer New York, 2003, pages 133-154.

Karim Belabas and Henri Cohen, Computation of the Hardy-Littlewood constant for quadratic polynomials, PARI/GP script, 2020.
Henri Cohen, High precision computation of Hardy-Littlewood constants. [Cached pdf version, with permission]
Keith Conrad, Hardy-Littlewood Constants, (2003).
Michael J. Jacobson Jr. and Hugh C. Williams, New Quadratic Polynomials With High Densities Of Prime Values, Math. Comp., 72, 241, 499-519, 2002.

Cf. A002837, A007635, A014556, A116206, A331877.

Cf. A221712, A221713 (Constants C including factor 1/2).

\\ The function HardyLittlewood2 is provided at the Belabas, Cohen link.
hl2max=0; for(add=0,100,my(hl=HardyLittlewood2(n^2+n+add)); if(hl>hl2max,print1(add,", "); hl2max=hl))

A128878 Primes of the form 47n^2 - 1701n + 10181.

Original entry on oeis.org

10181, 8527, 6967, 5501, 4129, 2851, 1667, 577, 379, 1451, 2617, 3877, 5231, 6679, 8221, 9857, 11587, 13411, 15329, 17341, 19447, 21647, 31387, 34057, 36821, 39679, 45677, 48817, 52051, 65927, 81307, 89561, 102647, 107197, 116579, 126337, 131357

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Apr 17 2007

nonn

Primes are given in the order in which they arise for increasing n.
Polynomial generates 22 primes for 0 <= n <= 42, i.e., for n = 0, 1, 2, 3, 4, 5, 6, 7, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42.
If the definition is replaced by "Numbers n of the form 47*k^2 - 1701*k + 10181 such that either n or -n is a prime" we get (essentially) A050267.

			47k^2 - 1701k + 10181 = 21647 for k = 42.

R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, ISBN 0-387-20860-7, Section A17, page 59.

G. W. Fung and H. C. Williams, Quadratic polynomials which have a high density of prime values, Math. Comput. 55(191) (1990), 345-353.
Carlos Rivera, Problem 12: Prime producing polynomials, The Prime Puzzles and Problems Connection.

Cf. A050267, A002383, A027753, A027755, A005471, A027758, A048059, A007635, A005846, A116206, A050268, A022464.

Select[Table[47*n^2 - 1701*n + 10181, {n, 0, 100}], # > 0 && PrimeQ[#] &] (* T. D. Noe, Aug 02 2011 *)

Edited by Klaus Brockhaus, Apr 22 2007 and by N. J. A. Sloane, May 05 2007 and May 06 2007

A253239 Numbers k such that k^2 + k + 72491 is prime.

Original entry on oeis.org

1, 2, 3, 6, 7, 11, 12, 13, 14, 16, 17, 19, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 53, 55, 56, 57, 58, 59, 64, 65, 66, 67, 72, 73, 74, 75, 77, 78, 81, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 98, 100

Offset: 1

Eric Chen, Apr 19 2015

Of the first 10000 natural numbers, 4534 are in this sequence, making the density about 45%, quite large! (However, 72491 is not prime; it equals 71*1021, so no multiples of 71 or 1021 are in this sequence.)

			k       k^2 + k + 72491
0       72491 = 71*1021
1       72493 (prime)
2       72497 (prime)
3       72503 (prime)
4       72511 = 59*1229
5       72521 = 47*1543
6       72533 (prime)
7       72547 (prime)
8       72563 = 149*487
9       72581 = 181*401
etc.

Eric Chen, Table of n, a(n) for n = 1..4534 (all terms up to 10000)
C. Rivera, Prime producing polynomials
Eric Weisstein's World of Mathematics, Prime-generating polynomial

Cf. A048097, A028823, A056561, A141489, A160548, A116206, A050267, A128878.

[n: n in [0..100] | IsPrime(n^2 + n + 72491)]; // Vincenzo Librandi, Apr 20 2015

select(t -> isprime(t^2+t+72491), [$0..100]);

Select[Range[100], PrimeQ[#^2 + # + 72491] &]

v=[ ]; for(n=0, 100, if(isprime(n^2+n+72491), v=concat(v, n), )); v

cp's OEIS Frontend

A050267 Primes or negative values of primes in the sequence b(n) = 47*n^2 - 1701*n + 10181, n >= 0.

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A331940 Addends k > 0 such that the polynomial x^2 + x + k produces a record of its Hardy-Littlewood Constant.

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PARI

A128878 Primes of the form 47*n^2 - 1701*n + 10181.

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Author

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Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

A253239 Numbers k such that k^2 + k + 72491 is prime.

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Mathematica

PARI

A050267 Primes or negative values of primes in the sequence b(n) = 47n^2 - 1701n + 10181, n >= 0.

A128878 Primes of the form 47n^2 - 1701n + 10181.