A005846 -id:A005846 - cp's OEIS frontend

A272438 Primes of the form abs(-66n^3 + 3845n^2 - 60897n + 251831) in order of increasing nonnegative n.

251831, 194713, 144889, 101963, 65539, 35221, 10613, 8681, 23057, 32911, 38639, 40637, 39301, 35027, 28211, 19249, 8537, 3529, 16553, 30139, 43891, 57413, 70309, 82183, 92639, 101281, 107713, 111539, 112363, 109789, 103421, 92863, 77719, 57593, 32089, 811

Offset: 1

Robert Price, Apr 29 2016

nonn

			65539 is in this sequence since abs(-66*4^3 + 3845*4^2 - 60897*4 + 251831) = abs(-4224+61520-243588+251831) = 65539 is prime.

Robert Price, Table of n, a(n) for n = 1..3012
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

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

Cf. A271980, A272030, A272074, A272075, A272159, A271143, A272284, A272302, A272437.

n = Range[0, 100]; Select[-66n^3 + 3845n^2 - 60897n + 251831, PrimeQ[#] &]

A272437 Nonnegative numbers n such that abs(-66n^3 + 3845n^2 - 60897n + 251831) 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, 47, 49, 51, 54, 58, 65, 68, 70, 75, 76, 77, 82, 88, 89, 97, 99, 101, 102, 104, 109

Offset: 1

Robert Price, Apr 29 2016

nonn

46 is the smallest number not in this sequence.

			4 is in this sequence since abs(-66*4^3 + 3845*4^2 - 60897*4 + 251831) = abs(-4224+61520-243588+251831) = 65539 is prime.

Robert Price, Table of n, a(n) for n = 1..3012
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A050268, A050267, A005846, A007641, A007635, A048988, A050265, A050266, A271980, A272030, A272074, A272075, A272160, A271144, A272285, A272401, A272438.

Select[Range[0, 109], PrimeQ[-66#^3 + 3845#^2 - 60897# + 251831] &]

is(n)=isprime(abs(66*n^3-3845*n^2+60897*n-251831)) \\ Charles R Greathouse IV, Feb 20 2017

A272444 Primes of the form abs(n^5 - 99n^4 + 3588n^3 - 56822n^2 + 348272n - 286397) in order of increasing nonnegative n.

Original entry on oeis.org

286397, 8543, 210011, 336121, 402851, 424163, 412123, 377021, 327491, 270631, 212123, 156353, 106531, 64811, 32411, 9733, 3517, 8209, 5669, 2441, 14243, 27763, 41051, 52301, 59971, 62903, 60443, 52561, 39971, 24251, 7963, 5227, 10429, 1409, 29531, 91673

Offset: 1

Robert Price, Apr 29 2016

nonn

			402851 is in this sequence since abs(4^5 - 99*4^4 + 3588*4^3 - 56822*4^2 + 348272*4 - 286397) = abs(1024-25344+229632-909152+1393088-286397) = 402851 is prime.

Robert Price, Table of n, a(n) for n = 1..2170
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

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

Cf. A271980, A272030, A272074, A272075, A272159, A271143, A272284, A272302, A272437, A272443.

n = Range[0, 100]; Select[n^5 - 99n^4 + 3588n^3 - 56822n^2 + 348272n - 286397, PrimeQ[#] &]

lista(nn) = for(n=0, nn, if(isprime(p=abs(n^5-99*n^4+3588*n^3-56822*n^2+348272*n-286397)), print1(p, ", "))); \\ Altug Alkan, Apr 29 2016

A057604 Primes of the form 4*k^2 + 163.

Original entry on oeis.org

163, 167, 179, 199, 227, 263, 307, 359, 419, 487, 563, 647, 739, 839, 947, 1063, 1187, 1319, 1459, 1607, 2099, 2467, 2663, 3079, 3299, 3527, 4007, 4259, 4519, 4787, 5347, 5639, 5939, 6247, 6563, 7219, 7559, 7907, 8263, 8627, 8999, 9767, 10163, 10567, 10979, 11399, 11827, 12263

Offset: 1

Tito Piezas III, Oct 08 2000

These numbers are not prime in O_Q(sqrt(-163)). If p = n^2 + 163, then (n - sqrt(-163))*(n + sqrt(-163)) = p. - Alonso del Arte, Dec 18 2017

Iain Fox, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)
S. A. Goudsmit, Unusual Prime Number Sequences, Nature Vol. 214 (1967), 1164.
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

Cf. A005846, A007641, A007635, A057605.

[a: n in [0..400] | IsPrime(a) where a is 4*n^2 + 163] // Vincenzo Librandi, Aug 07 2010

Select[Table[4n^2 + 163, {n, 0, 70}], PrimeQ] (* Vincenzo Librandi, Jul 15 2012 *)

lista(nn) = for(n=0, nn, my(p = 4*n^2 + 163); if(isprime(p), print1(p, ", "))) \\ Iain Fox, Dec 19 2017

Sequence corrected by Vincenzo Librandi, Jul 15 2012

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

A160548 Primes of the form k^2 + k + 844427.

Original entry on oeis.org

844427, 844429, 844433, 844439, 844447, 844457, 844469, 844483, 844499, 844517, 844609, 844733, 844769, 844847, 845027, 845129, 845183, 845357, 845833, 845909, 845987, 846067, 846149, 846233, 846407, 846589, 846779, 846877, 846977, 847079, 847507, 847967, 848087

Offset: 1

Arkadiusz Wesolowski, May 18 2009

nonn

844427 is the fourth term of A190800 and of A191456. - Arkadiusz Wesolowski, Jun 25 2011

Vincenzo Librandi and Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000 (the first 210 terms are from Vincenzo Librandi)
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 844427
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

Cf. A005846, A007635, A048059, A190800, A191456.

[n^2+n+844427 : n in [0..60] | IsPrime(n^2+n+844427)]; // Bruno Berselli, Feb 23 2011

Select[Table[n^2 + n + 844427, {n, 0, 60}], PrimeQ] (* Arkadiusz Wesolowski, Mar 04 2011 *)

for(n=0, 60, if(isprime(x=(n^2+n+844427)), print1(x, ", "))); \\ Arkadiusz Wesolowski, Mar 02 2011

select(isprime, vector(1000, n, n^2+n+844427)) \\ Charles R Greathouse IV, Feb 23 2011

A256585 Primes of the form 3n^2 + 39n + 37.

Original entry on oeis.org

37, 79, 127, 181, 241, 307, 379, 457, 541, 631, 727, 829, 937, 1051, 1171, 1297, 1429, 1567, 1861, 2017, 2179, 2347, 2521, 2887, 3079, 3691, 3907, 4129, 4357, 4591, 4831, 5077, 5851, 6121, 6397, 6679, 6967, 7561, 7867, 8179, 8821, 9151, 9829, 10177, 10531

Offset: 1

S. J. Vincent, Apr 02 2015

Primes of the form 6*m+1 such that 8*m + 121 is a square. - Bruno Berselli, Apr 18 2016

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

Cf. A271980, A005846, A007635, A007641, A048988, A050265 - A050268.

select(isprime, [3*k*(k+13)+37$k=0..100])[];  # Alois P. Heinz, Apr 16 2025

Select[(3 #^2 + 39 # + 37) & /@ Range[0, 100], PrimeQ] (* Robert Price, Apr 16 2025 *)

A272323 Nonnegative numbers n such that abs(82n^3 - 1228n^2 + 6130n - 5861) 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, 34, 37, 39, 41, 43, 47, 49, 50, 53, 54, 55, 59, 61, 63, 64, 67, 72, 73, 75, 76, 81, 84, 86, 87, 88, 89, 90, 92, 95, 97, 98, 102, 103, 104

Offset: 1

Robert Price, Apr 25 2016

nonn

32 is the smallest number not in this sequence.

			4 is in this sequence since 82*4^3 - 1228*4^2 + 6130*4 - 5861 = 5248-19648+24520-5861 = 4259 is prime.

Robert Price, Table of n, a(n) for n = 1..2659
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, A272324, A076808.

Select[Range[0, 100], PrimeQ[82#^3 - 1228#^2 + 6130# - 5861] &]

lista(nn) = for(n=0, nn, if(isprime(abs(82*n^3-1228*n^2+6130*n-5861)), print1(n, ", "))); \\ Altug Alkan, Apr 25 2016

A272410 Primes of the form abs(n^4 - 97n^3 + 3294n^2 - 45458n + 213589) in order of increasing nonnegative n.

Original entry on oeis.org

213589, 171329, 135089, 104323, 78509, 57149, 39769, 25919, 15173, 7129, 1409, 2341, 4451, 5227, 4951, 3881, 2251, 271, 1873, 4019, 6029, 7789, 9209, 10223, 10789, 10889, 10529, 9739, 8573, 7109, 5449, 3719, 2069, 673, 271, 541, 109, 1949, 5273, 10399, 17669

Offset: 1

Robert Price, Apr 30 2016

nonn

			78509 is in this sequence since abs(4^4 - 97*4^3 + 3294*4^2 - 45458*4 + 213589) = abs(256-6208+52704-181832+213589) = 78509 is prime.

Robert Price, Table of n, a(n) for n = 1..2676
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

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

Cf. A271980, A272030, A272074, A272075, A272159, A271143, A272284, A272302, A272437, A272443, A268200.

n = Range[0, 100]; Select[n^4 - 97n^3 + 3294n^2 - 45458n + 213589, PrimeQ[#] &]

A272443 Nonnegative numbers n such that abs(n^5 - 99n^4 + 3588n^3 - 56822n^2 + 348272n - 286397) 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, 48, 50, 51, 53, 57, 58, 59, 64, 67, 70, 75, 79, 80, 81, 89, 91, 92, 93, 96, 99

Offset: 1

Robert Price, Apr 29 2016

nonn

47 is the smallest number not in this sequence.

			4 is in this sequence since abs(4^5 - 99*4^4 + 3588*4^3 - 56822*4^2 + 348272*4 - 286397) = abs(1024-25344+229632-909152+1393088-286397) = 402851 is prime.

Robert Price, Table of n, a(n) for n = 1..2170
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, A272401, A272438, A272444.

Select[Range[0, 100], PrimeQ[#^5 - 99#^4 + 3588#^3 - 56822#^2 + 348272# - 286397] &]

lista(nn) = for(n=0, nn, if(isprime(abs(n^5-99*n^4+3588*n^3-56822*n^2+348272*n-286397)), print1(n, ", "))); \\ Altug Alkan, Apr 29 2016

cp's OEIS Frontend

A272438 Primes of the form abs(-66n^3 + 3845n^2 - 60897n + 251831) in order of increasing nonnegative n.

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A272437 Nonnegative numbers n such that abs(-66n^3 + 3845n^2 - 60897n + 251831) is prime.

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A272444 Primes of the form abs(n^5 - 99n^4 + 3588n^3 - 56822n^2 + 348272n - 286397) in order of increasing nonnegative n.

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A057604 Primes of the form 4*k^2 + 163.

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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.

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A160548 Primes of the form k^2 + k + 844427.

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A256585 Primes of the form 3n^2 + 39n + 37.

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A272323 Nonnegative numbers n such that abs(82n^3 - 1228n^2 + 6130n - 5861) is prime.

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.