A027758 -id:A027758 - 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

A027726 Numbers k such that k^2+k+9 is a palindrome.

Original entry on oeis.org

0, 1, 9, 11, 13, 22, 30, 31, 138, 300, 304, 305, 331, 438, 969, 1141, 1413, 2367, 3000, 3144, 3881, 9854, 30000, 30605, 72062, 106801, 114141, 125206, 128348, 300000, 315165, 963304, 980560, 989154, 2378507, 3000000, 3040604, 3045679, 3152290, 3932806

Offset: 1

Patrick De Geest

Giovanni Resta, Table of n, a(n) for n = 1..71
P. De Geest, Palindromic Quasi_Over_Squares of the form n^2+(n+X)

Cf. A027727, A027694, A027757, A027758, A027724.

palQ[n_] := Block[{d = IntegerDigits[n]}, d == Reverse[d]]; f[n_] := n^2 + n + 9; Select[Range[0, 3*10^5], palQ@ f@ # &]

More terms from Giovanni Resta, Aug 29 2018

A027727 Palindromes of form k^2 + k + 9.

Original entry on oeis.org

9, 11, 99, 141, 191, 515, 939, 1001, 19191, 90309, 92729, 93339, 109901, 192291, 939939, 1303031, 1997991, 5605065, 9003009, 9887889, 15066051, 97111179, 900030009, 936696639, 5193003915, 11406560411, 13028282031, 15676667651, 16473337461, 90000300009

Offset: 1

Patrick De Geest

Palindromes h such that 4*h - 35 is a square. - Bruno Berselli, Aug 29 2018

Giovanni Resta, Table of n, a(n) for n = 1..71
P. De Geest, Palindromic Quasi_Over_Squares of the form n^2+(n+X)

Cf. A027726, A027694, A027757, A027758, A027725.

palQ[n_] := Block[{d = IntegerDigits[n]}, d == Reverse[d]]; f[n_] := n^2 + n + 9; Select[f@ Range[0, 10^5], palQ] (* Giovanni Resta, Aug 29 2018 *)
Select[Table[k^2+k+9,{k,0,300000}],PalindromeQ] (* Harvey P. Dale, Nov 21 2024 *)

More terms from Giovanni Resta, Aug 29 2018

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

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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A027726 Numbers k such that k^2+k+9 is a palindrome.

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A027727 Palindromes of form k^2 + k + 9.

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

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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.