A022464 -id:A022464 - cp's OEIS frontend

A050268 Primes of the form 36k^2 - 810k + 2753, listed in order of increasing parameter k >= 0.

2753, 1979, 1277, 647, 89, 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, 41669, 44207, 46817, 49499, 52253

Offset: 1

Eric W. Weisstein

The sequence of primes of this form, in order of increasing size, would read: 89, 359, 647, 953, 1277, 1619, 1979, 2357, 2753, ... - M. F. Hasler, Jan 18 2015

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. In the present sequence only positive values of the polynomial are taken into account. A117081 provides also the negative function values. - Hugo Pfoertner, Dec 13 2019

Paulo Ribenboim, The Little Book of Bigger Primes, Second Edition, Springer-Verlag New York, 2004.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Jitender Singh, Prime numbers and factorization of polynomials, arXiv:2411.18366 [math.NT], 2024.
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.

Cf. A022464, A117081.

[a: n in [0..100] | IsPrime(a) where a is  36*n^2 - 810*n + 2753]; // Vincenzo Librandi, Dec 08 2011

t1:=[seq(36*n^2 - 810*n + 2753,n=0..100)]; t2:=[]; for i from 1 to nops(t1) do if isprime(t1[i]) then t2:=[op(t2),t1[i]]; fi; od: t2; # N. J. A. Sloane

Select[Table[36n^2-810n+2753,{n,0,2000}],PrimeQ] (* Vincenzo Librandi, Dec 08 2011 *)

select(isprime, vector(1000, n, 36*n^2-810*n+2753)) \\ Charles R Greathouse IV, Feb 14 2011

Definition corrected by M. F. Hasler, Jan 18 2015

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

Original entry on oeis.org

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

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

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A050268 Primes of the form 36k^2 - 810k + 2753, listed in order of increasing parameter k >= 0.

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

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

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cp's OEIS Frontend

A050268 Primes of the form 36*k^2 - 810*k + 2753, listed in order of increasing parameter k >= 0.

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Crossrefs

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Magma

Maple

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Extensions

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

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Crossrefs

Programs

Mathematica

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Extensions

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

A050268 Primes of the form 36k^2 - 810k + 2753, listed in order of increasing parameter k >= 0.

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.