A050266 -id:A050266 - cp's OEIS frontend

A272324 Primes of the form abs(82n^3 - 1228n^2 + 6130n - 5861) in order of increasing nonnegative n.

5861, 877, 2143, 3691, 4259, 4339, 4423, 5003, 6571, 9619, 14639, 22123, 32563, 46451, 64279, 86539, 113723, 146323, 184831, 229739, 281539, 340723, 407783, 483211, 567499, 661139, 764623, 878443, 1003091, 1139059, 1286839, 1446923, 2005919, 2693363, 3229579

Offset: 1

Robert Price, Apr 25 2016

nonn

			4259 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. A076808, A050268, A050267, A005846, A007641, A007635, A048988, A050265, A050266.

Cf. A271980, A272030, A272074, A272075, A272118, A272159, A271143, A272284, A272323.

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

A272325 Nonnegative numbers n such that n^4 + 853n^3 + 2636n^2 + 3536n + 1753 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, 22, 25, 26, 27, 30, 34, 37, 41, 43, 46, 50, 52, 53, 56, 59, 60, 61, 64, 66, 67, 68, 71, 76, 79, 81, 84, 87, 88, 89, 91, 92, 95, 96, 98, 99, 103, 106, 109, 118, 124, 126, 127, 128, 132

Offset: 1

Robert Price, Apr 25 2016

nonn

21 is the smallest number not in this sequence.

			4 is in this sequence since 4^4 + 853*4^3 + 2636*4^2 + 3536*4 + 1753 = 256+54592+42176+14144+1753 = 112921 is prime.

Robert Price, Table of n, a(n) for n = 1..2457
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, A272326, A076809.

Select[Range[0, 100], PrimeQ[#^4 + 853#^3 + 2636#^2 + 3536# + 1753] &]

lista(nn) = for(n=0, nn, if(isprime(n^4+853*n^3+2636*n^2+3536*n+1753), print1(n, ", "))); \\ Altug Alkan, Apr 25 2016

A272326 Primes of the form k^4 + 853k^3 + 2636k^2 + 3536*k + 1753 in order of increasing nonnegative k.

Original entry on oeis.org

1753, 8779, 26209, 59197, 112921, 192583, 303409, 450649, 639577, 875491, 1163713, 1509589, 1918489, 2395807, 2946961, 3577393, 4292569, 5097979, 5999137, 7001581, 8110873, 10672369, 15456403, 17324929, 19339909, 26321233, 38031841, 48822439, 66193219

Offset: 1

Robert Price, Apr 25 2016

nonn

			112921 is in this sequence since 4^4 + 853*4^3 + 2636*4^2 + 3536*4 + 1753 = 256+54592+42176+14144+1753 = 112921 is prime.

Robert Price, Table of n, a(n) for n = 1..2457
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, A272325, A076809.

n = Range[0, 100]; Select[n^4 + 853n^3 + 2636n^2 + 3536n + 1753, PrimeQ[#] &]

lista(nn) = for(n=0, nn, if(isprime(p=n^4+853*n^3+2636*n^2+3536*n+1753), print1(p, ", "))); \\ Altug Alkan, Apr 25 2016

A272554 Nonnegative numbers n such that abs(1/(36)(n^6 - 126n^5 + 6217n^4 - 153066n^3 + 1987786n^2 - 13055316n + 34747236)) 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, 57, 61, 62, 63, 64, 65, 66, 68, 69, 70, 73, 78

Offset: 1

Robert Price, May 02 2016

nonn

55 is the smallest number not in this sequence.

			4 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, A272160, A271144, A272285, A272401, A272438, A272444, A272410, A272555.

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

A272710 Primes of the form abs((1/4)*(n^5 - 133n^4 + 6729n^3 - 158379n^2 + 1720294n - 6823316)) in order of increasing nonnegative n.

Original entry on oeis.org

1705829, 1313701, 991127, 729173, 519643, 355049, 228581, 134077, 65993, 19373, 10181, 26539, 33073, 32687, 27847, 20611, 12659, 5323, 383, 3733, 4259, 1721, 3923, 12547, 23887, 37571, 53149, 70123, 87977, 106207, 124351, 142019, 158923, 174907, 189977

Offset: 1

Robert Price, May 04 2016

nonn

			519643 is in this sequence since abs(1/4 (n^5 - 133n^4 + 6729n^3 - 158379n^2 + 1720294n - 6823316)) = abs((1024 - 34048 + 430656 - 2534064 + 6881176 - 6823316)/4) = 519643 is prime.

Robert Price, Table of n, a(n) for n = 1..2328
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, A247163.

n = Range[0, 100]; Select[1/4 (n^5 - 133n^4 + 6729n^3 - 158379n^2 + 1720294n - 6823316), PrimeQ[#] &]

A046135 Primes p such that p+2 and p+12 are primes.

Original entry on oeis.org

5, 11, 17, 29, 41, 59, 71, 101, 137, 179, 227, 239, 269, 281, 347, 419, 431, 641, 809, 827, 1019, 1049, 1091, 1151, 1277, 1289, 1427, 1481, 1487, 1607, 1697, 1721, 1877, 2027, 2087, 2129, 2141, 2339, 2381, 2687, 2729, 2789, 2999, 3359, 3527, 3581

Offset: 1

Eric W. Weisstein

From Jonathan Vos Post, May 17 2006: (Start)
Could be defined as "Numbers n such that k^3+k^2+n is prime for k = 0, 1, 2."
The following subset is also prime for k = 3: 5, 11, 17, 71, 101, 137, 227, 281, 347, 431, 641, 827, 1151, 1277, 1487.  The following subset of those is also prime for k = 4: 17, 71, 101, 227, 827, 1151, 1487.  The following subset of those is also prime for k = 5: 827, 1151, 1487.  The "17" in A050266's n^3+n^2+17 is because k^3+k^2+17 is prime for k = 1,2,3,4,5,6,7,8,9,10. Between 10000 and 20000 there are 30 members of the k = 0,1,2 sequence, of which these 10 are also prime for k = 3: 10301, 10937, 11057, 11777, 12107, 13997, 15137, 15737, 16061, 19541.  The following subset of those is also prime for k = 5: 15137, 15737, 16061.  Somewhere in these sequences is a value that breaks the 11-term record of A050266 and indeed any known prime generating polynomial record. (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Prime Triplet

Cf. A000040, A001359, A046133, A050266.

[p: p in PrimesUpTo(3600) | IsPrime(p+2) and IsPrime(p+12)]; // Vincenzo Librandi, Apr 09 2013

Select[Prime[Range[600]], PrimeQ[# + 2] && PrimeQ[# + 12]&] (* Vincenzo Librandi, Apr 09 2013 *)
Select[Prime[Range[600]],AllTrue[#+{2,12},PrimeQ]&] (* Harvey P. Dale, Jun 26 2025 *)

{n such that n prime, n+2 prime, n+12 prime} = A001359 INTERSECT A046133. - Jonathan Vos Post, May 17 2006

Edited by R. J. Mathar and N. J. A. Sloane, Aug 13 2008

A105551 Number of distinct prime factors of n^3 + n^2 + 71.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 2, 1, 1, 1, 1, 3, 2, 2, 2, 2, 1, 3, 3, 2, 1, 2, 1, 1, 3, 2, 3, 1, 2, 1, 1, 3, 2, 1, 2, 3, 2, 2, 2, 1, 3, 1, 1, 3, 2, 1, 2

Offset: 0

Jonathan Vos Post, May 03 2005

This cubic equation with small positive coefficients is strangely rich in primes and semiprimes. The first 44 consecutive values, for n = 0, 1, 2, ..., 43, are all either prime (23 of them) or semiprime (21 of them), before the first 3-almost prime value is encountered.

			a(0) = 1 because 0^3 + 0^2 + 71 = 71 is prime.
a(1) = 1 because 1^3 + 1^2 + 71 = 73 is prime.
a(2) = 1 because 2^3 + 2^2 + 71 = 83 is prime.
a(3) = 1 because 3^3 + 3^2 + 71 = 107 is prime.
a(4) = 1 because 3^3 + 3^2 + 71 = 151 is prime.
a(5) = 2 because 3^3 + 3^2 + 71 = 221 = 13 * 17 is the first semiprime.
a(44) = 3 because 44^3 + 44^2 + 71 = 87191 = 13 * 19 * 353 is the first 3-almost prime for nonnegative integers n.

T. D. Noe, Table of n, a(n) for n = 0..1000
Ulrich Abel and Hartmut Siebert, Sequences with Large Numbers of Prime Values, Am. Math. Monthly 100, 167-169, 1993.
Robin Forman, Sequences with Many Primes, Amer. Math. Monthly 99, 548-557, 1992.
Betty Garrison, Polynomials with Large Numbers of Prime Values, Amer. Math. Monthly 97, 316-317, 1990.
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.

Cf. A000040, A001358, A005846, A007635, A007641, A048988, A050265, A050268, A050267, A050266.

Table[PrimeNu[n^3+n^2+71],{n,0,90}] (* Harvey P. Dale, Oct 09 2012 *)

a(n)=omega(n^3 + n^2 + 71) \\ Charles R Greathouse IV, Jan 31 2017

a(n) = A001221(n^3 + n^2 + 71).

More terms from Robert G. Wilson v, May 21 2005

A247163 Nonnegative numbers n such that abs(1/4 (n^5 - 133n^4 + 6729n^3 - 158379n^2 + 1720294n - 6823316)) 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, 59, 60, 61, 64, 67, 68, 69, 74, 75, 76

Offset: 1

Robert Price, May 04 2016

nonn

62 is the smallest number not in this sequence.

			4 is in this sequence since abs(1/4 (n^5 - 133n^4 + 6729n^3 - 158379n^2 + 1720294n - 6823316)) = abs((1024 - 34048 + 430656 - 2534064 + 6881176 - 6823316)/4) = 519643 is prime.

Robert Price, Table of n, a(n) for n = 1..2328
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, A272410, A272555, A272710.

Select[Range[0, 100], PrimeQ[1/4 (#^5 - 133#^4 + 6729#^3 - 158379#^2 + 1720294# - 6823316)] &]

A267069 Nonnegative numbers n such that abs(103n^2 - 4707n + 50383) 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, 44, 45, 47, 49, 50, 51, 52, 53, 54, 57, 59, 60, 61, 63, 64, 65, 66, 67, 69, 73, 74, 76, 77, 80

Offset: 1

Robert Price, Apr 28 2016

nonn

43 is the smallest number not in this sequence.

See A267252 for more information. - Hugo Pfoertner, Dec 13 2019

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

Robert Price, Table of n, a(n) for n = 1..3885
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, A267252.

Select[Range[0, 100], PrimeQ[103#^2 - 4707# + 50383 ] &]

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

Title corrected by Hugo Pfoertner, Dec 13 2019

A272076 Numbers n such that abs(7n^2 - 371n + 4871) 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, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 59, 61, 63, 65, 67, 68, 72, 73, 74, 75, 76, 77, 78

Offset: 1

Robert Price, Apr 19 2016

nonn

			4 is in this sequence since 7*4^2 - 371*4 + 4871 = 112-1484+4871 = 3499 is prime.

Robert Price, Table of n, a(n) for n = 1..3530
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

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

Cf. A271980, A272074, A272075, A272077.

Select[Range[0, 100], PrimeQ[7#^2 - 371# + 4871] &]

lista(nn) = for(n=0, nn, if(ispseudoprime(abs(7*n^2-371*n+4871)), print1(n, ", "))); \\ Altug Alkan, Apr 19 2016

cp's OEIS Frontend

A272324 Primes of the form abs(82n^3 - 1228n^2 + 6130n - 5861) in order of increasing nonnegative n.

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A272325 Nonnegative numbers n such that n^4 + 853n^3 + 2636n^2 + 3536n + 1753 is prime.

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A272326 Primes of the form k^4 + 853*k^3 + 2636*k^2 + 3536*k + 1753 in order of increasing nonnegative k.

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A272554 Nonnegative numbers n such that abs(1/(36)(n^6 - 126n^5 + 6217n^4 - 153066n^3 + 1987786n^2 - 13055316n + 34747236)) is prime.

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A272710 Primes of the form abs((1/4)*(n^5 - 133n^4 + 6729n^3 - 158379n^2 + 1720294n - 6823316)) in order of increasing nonnegative n.

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A046135 Primes p such that p+2 and p+12 are primes.

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A105551 Number of distinct prime factors of n^3 + n^2 + 71.

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A247163 Nonnegative numbers n such that abs(1/4 (n^5 - 133n^4 + 6729n^3 - 158379n^2 + 1720294n - 6823316)) is prime.

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A272326 Primes of the form k^4 + 853k^3 + 2636k^2 + 3536*k + 1753 in order of increasing nonnegative k.

A267069 Nonnegative numbers n such that abs(103n^2 - 4707n + 50383) is prime.

A272076 Numbers n such that abs(7n^2 - 371n + 4871) is prime.