A050268 -id:A050268 - cp's OEIS frontend

A115244 Indices of primes generated by Fung and Ruby's prime generating polynomial A050268.

402, 299, 206, 118, 24, 78, 141, 191, 224, 257, 272, 279, 276, 264, 242, 211, 167, 112, 38, 72, 162, 256, 350, 448, 558, 670, 786, 913, 1042, 1181, 1319, 1462, 1620, 1777, 1942, 2119, 2289, 2473, 2664, 2851, 3051, 3250, 3458, 3684

Offset: 0

Roger L. Bagula, May 11 2006

nonn

PrimePi[Negative Prime]=0 :Absolute value is necessary because of this.

G. C. Greubel, Table of n, a(n) for n = 0..7419

Cf. A005846, A050268, A117081.

map(numtheory:-pi,select(isprime, [seq(abs(36*n^2 - 810* n + 2753), n=0..300)])); # Robert Israel, Mar 03 2016

Table[PrimePi[Abs[36*n^2 - 810*n + 2753]], {n, 0, 43}]
a := Select[Table[36 n^2 - 810 n + 2753, {n, 0, 200}], PrimeQ];
PrimePi[Abs[a]] (* G. C. Greubel, Feb 08 2016 *)

a(n) = PrimePi(Abs(A050268(n))).

Name edited by Robert Israel, Mar 03 2016

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

A271980 Numbers k such that 3k^2 + 39k + 37 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, 19, 20, 21, 22, 23, 25, 26, 29, 30, 31, 32, 33, 34, 35, 38, 39, 40, 41, 42, 44, 45, 46, 48, 49, 51, 52, 53, 54, 55, 57, 58, 59, 60, 63, 64, 66, 68, 69, 70, 71, 72, 79, 84, 86, 88, 89, 90, 91, 92

Offset: 1

Robert Price, Apr 17 2016

From Peter Bala, Apr 16 2018: (Start)
Let P(n) = 3*n^2 + 39*n + 37. The absolute values of the polynomial P(2*n - 29) = 12*n^2 - 270*n + 1429 for n from 0 to 27 are distinct primes, except at n = 14 when the value is 1.
The absolute values of the polynomial 3*P((n - 20)/3) = n^2 - n - 269 for n from 0 to 42 are either prime or 3 times a prime.
The absolute values of the polynomial 3*P((4*n - 89)/3) = 16*n^2 - 556*n + 4561 for n from 0 to 27 are either prime or 3 times a prime. (End)

			4 is in this sequence since 3*4^2 + 39*4 + 37 = 48+156+37 = 241 is prime.

Robert Price, Table of n, a(n) for n = 1..3510
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A256585, A050265 - A050268, A005846, A007641, A007635, A048988.

[n: n in [0..100] |IsPrime(3*n^2+39*n+37)]; // Vincenzo Librandi, Apr 19 2018

Select[Range[0, 100], PrimeQ[3*#^2 + 39*# + 37] &]

isok(n) = isprime(3*n^2 + 39*n + 37); \\ Michel Marcus, Apr 17 2016

lista(nn) = for(n=0, nn, if(ispseudoprime(3*n^2+39*n+37), print1(n, ", "))); \\ Altug Alkan, Apr 18 2016

A272074 Numbers k such that k^4 + 29*k^2 + 101 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, 21, 23, 26, 31, 32, 34, 35, 37, 43, 44, 45, 47, 49, 53, 56, 60, 61, 62, 66, 67, 68, 70, 71, 72, 74, 75, 79, 80, 81, 84, 85, 89, 90, 91, 93, 96, 99

Offset: 1

Robert Price, Apr 19 2016

			4 is in this sequence since 4^4 + 29*4^2 + 101 = 256+464+101 = 821 is prime.

Robert Price, Table of n, a(n) for n = 1..2264
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A050265 - A050268, A005846, A007641, A007635, A048988, A256585.

Cf. A271980, A272075.

Select[Range[0,100],PrimeQ[#^4+29#^2+101]&] (* Harvey P. Dale, Dec 15 2020 *)

lista(nn) = for(n=0, nn, if(ispseudoprime(n^4+29*n^2+101), print1(n, ", "))); \\ Altug Alkan, Apr 19 2016

A272075 Primes of the form k^4 + 29*k^2 + 101.

Original entry on oeis.org

101, 131, 233, 443, 821, 1451, 2441, 3923, 6053, 9011, 13001, 18251, 25013, 33563, 44201, 57251, 73061, 92003, 114473, 140891, 207371, 295283, 476681, 951491, 1078373, 1369961, 1536251, 1913963, 3472523, 3804341, 4159451, 4943843, 5834531, 7972043, 9925541

Offset: 1

Robert Price, Apr 19 2016

			233 is prime and it is in this sequence since 233 = 2^4 + 29*2^2 + 101.

Robert Price, Table of n, a(n) for n = 1..2264
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

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

Cf. A271980, A272074.

n = Range[0, 100]; Select[#^4 + 29#^2 + 101, PrimeQ[#] &]

lista(nn) = for(n=0, nn, if(ispseudoprime(p=n^4+29*n^2+101), print1(p, ", "))); \\ Altug Alkan, Apr 19 2016

A272160 Primes of the form abs(8n^2 - 488n + 7243) in order of increasing nonnegative values of n.

Original entry on oeis.org

7243, 6763, 6299, 5851, 5419, 5003, 4603, 4219, 3851, 3499, 3163, 2843, 2539, 2251, 1979, 1723, 1483, 1259, 1051, 859, 683, 523, 379, 251, 139, 43, 37, 101, 149, 181, 197, 197, 181, 149, 101, 37, 43, 139, 251, 379, 523, 683, 859, 1051, 1259, 1483, 1723, 1979

Offset: 1

Robert Price, Apr 21 2016

nonn

			5419 is in this sequence since 8*4^2 - 488*4 + 7243 = 128-1952+7243 = 5419 is prime.

Robert Price, Table of n, a(n) for n = 1..4058
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

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

Cf. A271980, A272074, A272075, A272118, A272159.

n = Range[0, 100]; Select[Abs[8n^2 - 488n + 7243], PrimeQ[#] &]

lista(nn) = for(n=0, nn, if(isprime(p=abs(8*n^2-488*n+7243)), print1(p, ", "))); \\ Altug Alkan, Apr 21 2016

A271144 Primes of the form 42k^3 + 270k^2 - 26436*k + 250703 in order of increasing k.

Original entry on oeis.org

250703, 224579, 199247, 174959, 151967, 130523, 110879, 93287, 77999, 65267, 55343, 48479, 44927, 44939, 48767, 56663, 68879, 85667, 107279, 133967, 165983, 203579, 247007, 296519, 352367, 414803, 484079, 560447, 644159, 735467, 834623, 941879, 1057487

Offset: 1

Robert Price, Apr 23 2016

nonn

			151967 is prime and it is in this sequence since 151967 = 42*4^3 + 270*4^2 - 26436*4 + 250703.

Robert Price, Table of n, a(n) for n = 1..3092
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A050265 - A050268, A005846, A007641, A007635, A048988, A256585

Cf. A271980, A272074, A272075, A272118, A272159, A271143 (associated k).

n = Range[0, 100]; Select[42n^3 + 270n^2 - 26436n + 250703, PrimeQ[#] &]

A272159 Numbers k such that abs(8k^2 - 488k + 7243) 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, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 64, 65, 66, 67, 71

Offset: 1

Robert Price, Apr 21 2016

nonn

From Robert Israel, Apr 21 2016: (Start)
n such that either n <= 61 or 8n^2 - 488n + 7243 is prime.
The first number not in the sequence is 62. (End)

			4 is in this sequence since 8*4^2 - 488*4 + 7243 = 128-1952+7243 = 5419 is prime.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

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

Cf. A271980, A272074, A272075, A272160.

select(n -> isprime(abs(8*n^2 - 488*n + 7243)), [$0..1000]); # Robert Israel, Apr 21 2016

Select[Range[0, 100], PrimeQ[8#^2 - 488# + 7243] &]

lista(nn) = for(n=0, nn, if(isprime(abs(8*n^2-488*n+7243)), print1(n, ", "))); \\ Altug Alkan, Apr 21 2016

A271143 Numbers k such that 42k^3 + 270k^2 - 26436*k + 250703 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, 41, 43, 44, 48, 51, 54, 55, 56, 58, 61, 62, 63, 64, 65, 66, 67, 69, 71, 76, 78, 79, 84, 87, 88, 89, 90, 92

Offset: 1

Robert Price, Apr 23 2016

nonn

40 is the first value not in the sequence.

			4 is in this sequence since 42*4^3 + 270*4^2 - 26436*4 + 250703 = 151967, which is prime.

Robert Price, Table of n, a(n) for n = 1..3092
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A050265-A050268, A005846, A007641, A007635, A048988, A256585, A271980, A272074, A272075, A272160, A271144.

Select[Range[0, 100], PrimeQ[42#^3 + 270#^2 - 26436# + 250703] &]

is(n)=isprime(42*n^3+270*n^2-26436*n+250703) \\ Charles R Greathouse IV, Feb 17 2017

A272285 Primes of the form 43n^2 - 537n + 2971 in order of increasing nonnegative values of n.

Original entry on oeis.org

2971, 2477, 2069, 1747, 1511, 1361, 1297, 1319, 1427, 1621, 1901, 2267, 2719, 3257, 3881, 4591, 5387, 6269, 7237, 8291, 9431, 10657, 11969, 13367, 14851, 16421, 18077, 19819, 21647, 23561, 25561, 27647, 29819, 32077, 34421, 39367, 41969, 44657, 47431, 50291

Offset: 1

Robert Price, Apr 24 2016

			1511 is in this sequence since 43*4^2 - 537*4 + 2971 = 688-2148+2971 = 1511 is prime.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

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

Cf. A271980, A272030, A272074, A272075, A272118, A272159, A271143, A272284.

n = Range[0, 100]; Select[43n^2 - 537n + 2971, PrimeQ[#] &]

lista(nn) = for(n=0, nn, if(ispseudoprime(p=43*n^2 - 537*n + 2971), print1(p, ", "))); \\ Altug Alkan, Apr 24 2016

cp's OEIS Frontend

A115244 Indices of primes generated by Fung and Ruby's prime generating polynomial A050268.

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

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A271980 Numbers k such that 3*k^2 + 39*k + 37 is prime.

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A272074 Numbers k such that k^4 + 29*k^2 + 101 is prime.

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A272075 Primes of the form k^4 + 29*k^2 + 101.

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A272160 Primes of the form abs(8n^2 - 488n + 7243) in order of increasing nonnegative values of n.

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A271144 Primes of the form 42*k^3 + 270*k^2 - 26436*k + 250703 in order of increasing k.

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A272159 Numbers k such that abs(8*k^2 - 488*k + 7243) is prime.

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

A271980 Numbers k such that 3k^2 + 39k + 37 is prime.

A271144 Primes of the form 42k^3 + 270k^2 - 26436*k + 250703 in order of increasing k.

A272159 Numbers k such that abs(8k^2 - 488k + 7243) is prime.

A271143 Numbers k such that 42k^3 + 270k^2 - 26436*k + 250703 is prime.

A272285 Primes of the form 43n^2 - 537n + 2971 in order of increasing nonnegative values of n.