A267252 -id:A267252 - 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

A117081 a(n) = 36n^2 - 810n + 2753, producing the conjectured record number of 45 primes in a contiguous range of n for quadratic polynomials, i.e., abs(a(n)) is prime for 0 <= n < 44.

Original entry on oeis.org

2753, 1979, 1277, 647, 89, -397, -811, -1153, -1423, -1621, -1747, -1801, -1783, -1693, -1531, -1297, -991, -613, -163, 359, 953, 1619, 2357, 3167, 4049, 5003, 6029, 7127, 8297, 9539, 10853, 12239, 13697, 15227, 16829, 18503, 20249, 22067, 23957, 25919, 27953, 30059, 32237, 34487, 36809, 39203, 41669

Offset: 0

Roger L. Bagula, Apr 17 2006

The absolute values of a(n) for 0 <= n <= 44 are primes, a(45) = 39203 = 197*199. The positive prime terms are in A050268.

The polynomial is a transformed version of the polynomial P(x) = 36*x^2 + 18*x - 1801 whose absolute value gives 45 distinct primes for -33 <= x <= 11, found by Ruby in 1989. It is one of the 3 known quadratic polynomials whose absolute value produces more than 40 primes in a contiguous range from 0 to n. For the other two polynomials, which produce 43 primes, see A050267 and A267252. - Hugo Pfoertner, Dec 13 2019

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 = 0..1000
François Dress and Michel Olivier, Polynômes prenant des valeurs premières, Experimental Mathematics, Volume 8, Issue 4 (1999), pages 319-338.
Carlos Rivera, Problem 12: Prime producing polynomials, The Prime Puzzles and Problems Connection.
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005846, A050267, A050268, A117081, A267252.

I:=[2753, 1979, 1277]; [n le 3 select I[n] else 3*Self(n-1)-3 *Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, May 12 2012

f[n_] := If[Mod[n, 2] == 1, 36*n^2 - 810*n + 2753, 36*n^2 - 810*n + 2753] a = Table[f[n], {n, 0, 100}]
CoefficientList[Series[(2753-6280*x+3599*x^2)/(1-x)^3,{x,0,50}],x] (* Vincenzo Librandi, May 12 2012 *)
Table[36n^2-810n+2753,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{2753,1979,1277},50] (* Harvey P. Dale, Jun 20 2013 *)

{for(n=0, 46, print1(36*n^2-810*n+2753, ","))}

G.f.: (2753 - 6280*x + 3599*x^2)/(1-x)^3. - Colin Barker, May 10 2012
a(0)=2753, a(1)=1979, a(2)=1277, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jun 20 2013
E.g.f.: exp(x)*(2753 - 774*x + 36*x^2). - Elmo R. Oliveira, Feb 09 2025

Edited by N. J. A. Sloane, Apr 27 2007

Title extended by Hugo Pfoertner, Dec 13 2019

A267069 Nonnegative numbers n such that abs(103n^2 - 4707n + 50383) is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 47, 49, 50, 51, 52, 53, 54, 57, 59, 60, 61, 63, 64, 65, 66, 67, 69, 73, 74, 76, 77, 80

Offset: 1

Robert Price, Apr 28 2016

nonn

43 is the smallest number not in this sequence.

See A267252 for more information. - Hugo Pfoertner, Dec 13 2019

			4 is in this sequence since 103*4^2 - 4707*4 + 50383  = 1648-18828+50383 = 33203 is prime.

Robert Price, Table of n, a(n) for n = 1..3885
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A050268, A050267, A005846, A007641, A007635, A048988, A050265, A050266.

Cf. A271980, A272030, A272074, A272075, A272160, A271144, A272285, A267252.

Select[Range[0, 100], PrimeQ[103#^2 - 4707# + 50383 ] &]

lista(nn) = for(n=0, nn, if(isprime(abs(103*n^2-4707*n+50383)), print1(n, ", "))); \\ Altug Alkan, Apr 28 2016, corrected by Hugo Pfoertner, Dec 13 2019

Title corrected by Hugo Pfoertner, Dec 13 2019

cp's OEIS Frontend

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

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A117081 a(n) = 36n^2 - 810n + 2753, producing the conjectured record number of 45 primes in a contiguous range of n for quadratic polynomials, i.e., abs(a(n)) is prime for 0 <= n < 44.

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A267069 Nonnegative numbers n such that abs(103n^2 - 4707n + 50383) is prime.

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A117081 a(n) = 36*n^2 - 810*n + 2753, producing the conjectured record number of 45 primes in a contiguous range of n for quadratic polynomials, i.e., abs(a(n)) is prime for 0 <= n < 44.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A267069 Nonnegative numbers n such that abs(103*n^2 - 4707*n + 50383) is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

A117081 a(n) = 36n^2 - 810n + 2753, producing the conjectured record number of 45 primes in a contiguous range of n for quadratic polynomials, i.e., abs(a(n)) is prime for 0 <= n < 44.

A267069 Nonnegative numbers n such that abs(103n^2 - 4707n + 50383) is prime.