A007635 -id:A007635 - cp's OEIS frontend

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

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

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

A028823 Numbers k such that k^2 + k + 17 is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 35, 37, 38, 40, 42, 44, 45, 46, 47, 49, 53, 56, 57, 59, 60, 62, 63, 64, 70, 72, 73, 75, 76, 79, 81, 82, 86, 87, 91, 92, 95, 98, 103, 104, 108, 109, 110, 113, 114

Offset: 1

Patrick De Geest

nonn

Complement of A007636. - Michel Marcus, Jun 17 2013

			15^2 + 15 + 17 = 257, which is prime, so 15 is in the sequence.
16^2 + 16 + 17 = 289 = 17^2, so 16 is not in the sequence. Much more obviously, 17 is not in the sequence either.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Patrick De Geest, Palindromic Quasi_Over_Squares of the form n^2+(n+X)

Cf. A007635, A007636, A014556.

[n: n in [0..1000] |IsPrime(n^2+n+17)] // Vincenzo Librandi, Nov 19 2010

Select[Range[0, 199], PrimeQ[#^2 + # + 17] &] (* Indranil Ghosh, Mar 19 2017 *)

is(n)=isprime(n^2+n+17) \\ Charles R Greathouse IV, Feb 20 2017

from sympy import isprime
print([n for n in range(201) if isprime(n**2 + n + 17)]) # Indranil Ghosh, Mar 19 2017

A267252 Primes of the form abs(103n^2 - 4707n + 50383) in order of increasing nonnegative n.

Original entry on oeis.org

50383, 45779, 41381, 37189, 33203, 29423, 25849, 22481, 19319, 16363, 13613, 11069, 8731, 6599, 4673, 2953, 1439, 131, 971, 1867, 2557, 3041, 3319, 3391, 3257, 2917, 2371, 1619, 661, 503, 1873, 3449, 5231, 7219, 9413, 11813, 14419, 17231, 20249, 23473, 26903

Offset: 1

Robert Price, Apr 28 2016

This polynomial is a transformed version of the polynomial P(x) = 103*x^2 + 31*x - 3391 whose absolute value gives 43 distinct primes for -23 <= x <= 19, found by G. W. Fung in 1988. - Hugo Pfoertner, Dec 13 2019

			33203 is in this sequence since 103*4^2 - 4707*4 + 50383  = 1648-18828+50383 = 33203 is prime.

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

Robert Price, Table of n, a(n) for n = 1..3885
François Dress and Michel Olivier, Polynômes prenant des valeurs premières, Experimental Mathematics, Volume 8, Issue 4 (1999), 319-338.
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, A272323, A267069, A330363.

n = Range[0, 100]; Abs @ Select[103n^2 - 4707n + 50383 , PrimeQ[#] &]

lista(nn) = for(n=0, nn, if(isprime(p=abs(103*n^2-4707*n+50383)), print1(p, ", "))); \\ Altug Alkan, Apr 28 2016, corrected by Hugo Pfoertner, Dec 13 2019

Title corrected by Hugo Pfoertner, Dec 13 2019

A268200 Nonnegative numbers n such that abs(n^4 - 97n^3 + 3294n^2 - 45458n + 213589) 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, 51, 52, 53, 54, 62, 65, 67, 70, 72, 73, 74, 75, 84, 85, 86, 90, 92

Offset: 1

Robert Price, Apr 30 2016

nonn

50 is the smallest number not in this sequence.

			4 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, A271980, A272030, A272074, A272075, A272160, A271144, A272285, A272401, A272438, A272444, A272410.

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

is(n)=isprime(abs(n^4-97*n^3+3294*n^2-45458*n+213589)) \\ Charles R Greathouse IV, Feb 20 2017

A272555 Primes of the form abs(1/(36)(n^6 - 126n^5 + 6217n^4 - 153066n^3 + 1987786n^2 - 13055316n + 34747236)) in order of increasing nonnegative n.

Original entry on oeis.org

965201, 653687, 429409, 272563, 166693, 98321, 56597, 32969, 20873, 15443, 13241, 12007, 10429, 7933, 4493, 461, 3583, 6961, 9007, 9157, 7019, 2423, 4549, 13553, 23993, 35051, 45737, 54959, 61613, 64693, 63421, 57397, 46769, 32423, 16193, 1091, 8443, 6271

Offset: 1

Robert Price, May 02 2016

nonn

			166693 is in this sequence since abs(1/(36)(4^6 - 126*4^5 + 6217*4^4 - 153066*4^3 + 1987786*4^2 - 13055316*4 + 34747236)) = abs((4096 - 129024 + 1591552 - 9796224 + 31804576 - 5222126 + 34747236)/36) = 166693 is prime.

Robert Price, Table of n, a(n) for n = 1..2128
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, A272554.

n = Range[0, 100]; Select[1/(36)(n^6 - 126n^5 + 6217n^4 - 153066n^3 + 1987786n^2 - 13055316n + 34747236), PrimeQ[#] &]

cp's OEIS Frontend

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

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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.

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

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A272443 Nonnegative numbers n such that abs(n^5 - 99n^4 + 3588n^3 - 56822n^2 + 348272n - 286397) is prime.

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A028823 Numbers k such that k^2 + k + 17 is prime.

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A267252 Primes of the form abs(103*n^2 - 4707*n + 50383) in order of increasing nonnegative n.

A267252 Primes of the form abs(103n^2 - 4707n + 50383) in order of increasing nonnegative n.