A027755 -id:A027755 - 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

A027718 Numbers k such that k^2+k+5 is a palindrome.

Original entry on oeis.org

0, 1, 2, 8, 12, 26, 74, 127, 224, 230, 2751, 3462, 4012, 4067, 12752, 22424, 27548, 28168, 105322, 107422, 2358150, 2724718, 2775383, 4063892, 7569245, 85125933, 87579753, 106617617, 2237334999, 2426472519, 2765569146, 2781875716, 2815069131, 4029203527

Offset: 1

Patrick De Geest

Giovanni Resta, Table of n, a(n) for n = 1..46

Cf. A027728, A027690, A027754, A027755, A027716, A027729.

palQ[n_] := Block[{d = IntegerDigits[n]}, d == Reverse[d]]; f[n_] := n^2 + n + 5; Select[Range[0, 10^5], palQ@ f@ # &] (* Giovanni Resta, Aug 29 2018 *)

More terms from Giovanni Resta, Aug 28 2018

A027728 Palindromes of form k^2 + k + 5.

Original entry on oeis.org

5, 7, 11, 77, 161, 707, 5555, 16261, 50405, 53135, 7570757, 11988911, 16100161, 16544561, 162626261, 502858205, 758919857, 793464397, 11092829011, 11539593511, 5560873780655, 7424090904247, 7702753572077, 16515222251561, 57293477439275, 7246424554246427

Offset: 1

Patrick De Geest

Giovanni Resta, Table of n, a(n) for n = 1..46
P. De Geest, Palindromic Quasi_Over_Squares of the form n^2+(n+X)

Cf. A027718, A027690, A027754, A027755, A027717, A027728, A027721.

palQ[n_] := Block[{d = IntegerDigits[n]}, d == Reverse[d]]; f[n_] := n^2 + n + 5; Select[f@ Range[0, 10^5], palQ] (* Giovanni Resta, Aug 29 2018 *)
Select[Table[k^2+k+5,{k,0,852*10^5}],PalindromeQ] (* Harvey P. Dale, Aug 04 2025 *)

More terms from Giovanni Resta, Aug 28 2018

A027754 Numbers k such that k^2 + k + 5 is prime.

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 11, 16, 17, 18, 21, 23, 27, 31, 32, 38, 42, 48, 51, 62, 67, 72, 73, 76, 77, 83, 86, 91, 93, 97, 108, 111, 116, 121, 126, 133, 136, 137, 146, 153, 158, 163, 172, 177, 182, 188, 191, 192, 193, 202, 212, 213, 216, 223, 226, 231, 247, 248

Offset: 1

Patrick De Geest

nonn

			Since 3^2 + 3 + 5 = 17, which is prime, 3 is in the sequence.
Since 4^2 + 4 + 5 = 25 = 5^2, 4 is not in the sequence.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Patrick De Geest, World!Of Numbers

Cf. A014556, A027755.

[n: n in [0..1000] |IsPrime(n^2+n+5)] // Vincenzo Librandi, Nov 20 2010

Select[Range[0, 499], PrimeQ[#^2 + # + 5] &] (* Alonso del Arte, Nov 28 2016 *)

A302445 Triangle read by rows: row n gives primes of form k^2 + n - k for 0 < k < n.

Original entry on oeis.org

2, 3, 5, 5, 7, 11, 17, 7, 13, 19, 37, 11, 29, 11, 13, 17, 23, 31, 41, 53, 67, 83, 101, 13, 19, 43, 103, 17, 71, 197, 17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257, 19, 31, 61, 109, 151, 229, 23, 41, 131, 293, 401, 23, 29, 43, 53, 79, 113, 179, 233, 263, 443

Offset: 2

Seiichi Manyama, Apr 08 2018

			  n\k|  1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16
  ---+-----------------------------------------------------------------------
    2|  2;
    3|  3,  5;
    4|
    5|  5,  7, 11, 17;
    6|
    7|  7,   , 13, 19,   , 37;
    8|
    9|   , 11,   ,   , 29,   ,   ,   ;
   10|
   11| 11, 13, 17, 23, 31, 41, 53, 67, 83, 101;
   12|
   13| 13,   , 19,   ,   , 43,   ,   ,   , 103,    ,    ;
   14|
   15|   , 17,   ,   ,   ,   ,   , 71,   ,    ,    ,    ,    , 197;
   16|
   17| 17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257;

Seiichi Manyama, Rows n = 2..421, flattened
Eric Weisstein's World of Mathematics, Lucky Number of Euler
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial
Wikipedia, Lucky numbers of Euler

Row n: A027753 (n=3), A027755 (n=5), A048059 (n=11), A007635 (n=17), A005846 (n=41).

Cf. A000040, A014556, A302826.

a:=Filtered(Flat(List([1..10],n->List([1..n],k->k^2+n-k))),IsPrime); # Muniru A Asiru, Apr 09 2018

Map[Union@ Select[#, PrimeQ] &, Table[k^2 + n - k, {n, 23}, {k, 0, n}]] // Flatten (* Michael De Vlieger, Apr 10 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

A268101 Smallest prime p such that some polynomial of the form ax^2 - bx + p generates distinct primes in absolute value for x = 1 to n, where 0 < a < p and 0 <= b < p.

Original entry on oeis.org

2, 3, 5, 5, 7, 7, 11, 11, 11, 11, 13, 13, 17, 17, 17, 17, 19, 19, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 31, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 647, 1277, 1979, 2753

Offset: 1

Arkadiusz Wesolowski, Jan 26 2016

			a(1) = 2 (a prime), x^2 + 2 gives a prime for x = 1.
a(2) = 3 (a prime), 2*x^2 + 3 gives distinct primes for x = 1 to 2.
a(4) = 5 (a prime), 2*x^2 + 5 gives distinct primes for x = 1 to 4.
a(6) = 7 (a prime), 4*x^2 + 7 gives distinct primes for x = 1 to 6.
a(10) = 11 (a prime), 2*x^2 + 11 gives distinct primes for x = 1 to 10.
a(12) = 13 (a prime), 6*x^2 + 13 gives distinct primes for x = 1 to 12.
a(16) = 17 (a prime), 6*x^2 + 17 gives distinct primes for x = 1 to 16.
a(18) = 19 (a prime), 10*x^2 + 19 gives distinct primes for x = 1 to 18.
a(22) = 23 (a prime), 3*x^2 - 3*x + 23 gives distinct primes for x = 1 to 22.
a(28) = 29 (a prime), 2*x^2 + 29 gives distinct primes for x = 1 to 28.
a(29) = 31 (a prime), 2*x^2 - 4*x + 31 gives distinct primes for x = 1 to 29.
a(40) = 41 (a prime), x^2 - x + 41 gives distinct primes for x = 1 to 40.
a(41) = 647 (a prime), abs(36*x^2 - 594*x + 647) gives distinct primes for x = 1 to 41.
a(42) = 1277 (a prime), abs(36*x^2 - 666*x + 1277) gives distinct primes for x = 1 to 42.
a(43) = 1979 (a prime), abs(36*x^2 - 738*x + 1979) gives distinct primes for x = 1 to 43.
a(44) = 2753 (a prime), abs(36*x^2 - 810*x + 2753) gives distinct primes for x = 1 to 44.

Carlos Rivera, Problem 12
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

Cf. A027688, A027753, A027690, A027755, A048058, A048059, A007635, A007639, A007637, A007641, A202018, A005846, A117081, A050268, A268109.

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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A027718 Numbers k such that k^2+k+5 is a palindrome.

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A027728 Palindromes of form k^2 + k + 5.

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A027754 Numbers k such that k^2 + k + 5 is prime.

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A302445 Triangle read by rows: row n gives primes of form k^2 + n - k for 0 < k < n.

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GAP

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

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A268101 Smallest prime p such that some polynomial of the form a*x^2 - b*x + p generates distinct primes in absolute value for x = 1 to n, where 0 < a < p and 0 <= b < p.

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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.

A268101 Smallest prime p such that some polynomial of the form ax^2 - bx + p generates distinct primes in absolute value for x = 1 to n, where 0 < a < p and 0 <= b < p.