A076808 -id:A076808 - cp's OEIS frontend

A076809 a(n) = n^4 + 853n^3 + 2636n^2 + 3536n + 1753.

1753, 8779, 26209, 59197, 112921, 192583, 303409, 450649, 639577, 875491, 1163713, 1509589, 1918489, 2395807, 2946961, 3577393, 4292569, 5097979, 5999137, 7001581, 8110873, 9332599, 10672369, 12135817, 13728601, 15456403, 17324929, 19339909, 21507097, 23832271, 26321233, 28979809, 31813849, 34829227

Offset: 0

Hilko Koning (hilko(AT)hilko.net), Nov 18 2002

A prime-generating quartic polynomial.

For n=0 ... 20, the terms in this sequence are primes. This is not the case for n=21. See A272325 and A272326. - Robert Price, Apr 25 2016

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A076808, A272325, A272326.

A076809:=n->n^4 + 853*n^3 + 2636*n^2 + 3536*n + 1753; seq(A076809(n), n=0..100); # Wesley Ivan Hurt, Nov 13 2013

Table[n^4 + 853n^3 + 2636n^2 + 3536n + 1753, {n,0,100}] (* Wesley Ivan Hurt, Nov 13 2013 *)
CoefficientList[Series[-(x^4 - 1588 x^3 - 156 x^2 + 14 x + 1753)/(x - 1)^5, {x, 0, 33}], x] (* Michael De Vlieger, Apr 25 2016 *)
LinearRecurrence[{5,-10,10,-5,1},{1753,8779,26209,59197,112921},40] (* Harvey P. Dale, Jan 20 2025 *)

A076809(n):=n^4 + 853*n^3 + 2636*n^2 + 3536*n + 1753$
makelist(A076809(n),n,0,30); /* Martin Ettl, Nov 08 2012 */

G.f.: -(x^4-1588*x^3-156*x^2+14*x+1753)/(x- 1)^5. [Colin Barker, Nov 11 2012]

E.g.f.: (1753 + 7026*x + 5202*x^2 + 859*x^3 + x^4)*exp(x). - Ilya Gutkovskiy, Apr 25 2016

More terms from Michael De Vlieger, Apr 25 2016

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

A095946 Primes of the form 8n^4 - 2n^3 - 132n^2 + 318n + 80557.

Original entry on oeis.org

80557, 80749, 80777, 80917, 81637, 83597, 87649, 94837, 106397, 123757, 148537, 182549, 227797, 286477, 360977, 453877, 567949, 706157, 871657, 1067797, 1298117, 1566349, 1876417, 2232437, 2638717, 3099757, 3620249, 4205077, 4859317

Offset: 1

Hilko Koning (hilko(AT)hilko.net), Jul 13 2004

The first composite has n = 29.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein, Prime-Generating Polynomials

Cf. A076808, A076809.

[a: n in [0..100] | IsPrime(a) where a is 8*n^4-2*n^3-132*n^2+ 318*n+80557]; // Vincenzo Librandi, Jul 17 2012

Select[Table[8n^4-2n^3-132n^2+318n+80557,{n,0,2000}],PrimeQ] (* Vincenzo Librandi, Jul 17 2012 *)

A218456 2n^3 - 313n^2 + 6823*n - 13633.

Original entry on oeis.org

-13633, -7121, -1223, 4073, 8779, 12907, 16469, 19477, 21943, 23879, 25297, 26209, 26627, 26563, 26029, 25037, 23599, 21727, 19433, 16729, 13627, 10139, 6277, 2053, -2521, -7433, -12671, -18223, -24077, -30221, -36643, -43331, -50273, -57457

Offset: 0

Pedja Terzic, Oct 29 2012

A prime-producing cubic polynomial. Produces 79 distinct primes if we scan the absolute values of the first 100 terms..

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A124363, A076808.

Table[2n^3-313n^2+6823n-13633,{n,0,99}]
LinearRecurrence[{4,-6,4,-1},{-13633,-7121,-1223,4073},40] (* Harvey P. Dale, May 03 2018 *)

A218456(n):=2*n^3-313*n^2+6823*n-13633$
makelist(A218456(n),n,0,30); /* Martin Ettl, Nov 08 2012 */

G.f.: (20771*x^3-54537*x^2+47411*x-13633)/(x-1)^4. [Colin Barker, Nov 10 2012]

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

Original entry on oeis.org

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[#] &]

A218457 a(n) = 6n^3 - 263n^2 + 3469*n - 12841.

Original entry on oeis.org

-12841, -9629, -6907, -4639, -2789, -1321, -199, 613, 1151, 1451, 1549, 1481, 1283, 991, 641, 269, -89, -397, -619, -719, -661, -409, 73, 821, 1871, 3259, 5021, 7193, 9811, 12911, 16529, 20701, 25463, 30851, 36901, 43649, 51131, 59383, 68441, 78341, 89119

Offset: 0

Pedja Terzic, Oct 29 2012

A prime-producing cubic polynomial. Produces 78 distinct primes if we scan the absolute values of the first 100 terms.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A124363, A076808, A218456.

Mathematica
```
Table[6n^3-263n^2+3469n-12841,{n,0,99}]
```

a(n) = {6*n^3 - 263*n^2 + 3469*n - 12841} \\ Andrew Howroyd, Apr 27 2020

Signs of terms corrected and a(32) and beyond from Andrew Howroyd, Apr 27 2020

A095974 Evaluate n^4 - 93n^3 + 3196n^2 - 48008n + 265483 for n >= 0, record the primes.

Original entry on oeis.org

265483, 220579, 181523, 147793, 118891, 94343, 73699, 56533, 42443, 31051, 22003, 14969, 9643, 5743, 3011, 1213, 139, -397, -557, -479, -277, -41, 163, 293, 331, 283, 179, 73, 43, 191, 643, 1549, 3083, 5443

Offset: 0

Hilko Koning (hilko(AT)hilko.net), Jul 16 2004

A prime-generating quartic polynomial.

Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A076808, A076809, A095946.

A096372 List of primes produced by a certain "prime-generating" quartic polynomial.

Original entry on oeis.org

1688311, 1410743, 1168619, 958807, 778319, 624311, 494083, 385079, 294887, 221239, 162011, 115223, 79039, 51767, 31859, 17911, 8663, 2999, -53, -1321, -1489, -1097, -541, -73, 199, 311, 443, 919, 2207, 4919, 9811, 17783, 29879, 47287, 71339, 103511, 145423, 198839

Offset: 1

Hilko Koning (hilko(AT)hilko.net), Jul 19 2004

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

Cf. A076808, A076809, A095946, A095974.

f:= n -> 6*n^4 - 558*n^3 + 19354*n^2 - 296370*n + 1688311:
select(isprime@abs, [seq(f(n),n=0..100)]); # Robert Israel, Jan 16 2018

Select[Table[6n^4-558n^3+19354n^2-296370n+1688311,{n,0,40}],PrimeQ] (* Harvey P. Dale, Dec 31 2018 *)

Evaluate 6n^4 - 558n^3 + 19354n^2 - 296370n + 1688311 for n >= 0, record the primes.

Offset changed to 1 by Robert Israel, Jan 16 2018

A218458 a(n) = 2n^3 - 163n^2 + 2777*n - 11927.

Original entry on oeis.org

-11927, -9311, -7009, -5009, -3299, -1867, -701, 211, 881, 1321, 1543, 1559, 1381, 1021, 491, -197, -1031, -1999, -3089, -4289, -5587, -6971, -8429, -9949, -11519, -13127, -14761, -16409, -18059, -19699, -21317, -22901, -24439, -25919

Offset: 0

Pedja Terzic, Oct 29 2012

A prime-producing cubic polynomial. Produces 78 distinct primes if we scan the absolute values of the first 100 terms.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A124363, A076808, A218456, A218457.

[2*n^3 - 163*n^2 + 2777*n - 11927 : n in [0..60]]; // Wesley Ivan Hurt, Apr 21 2021

Table[2n^3-163n^2+2777n-11927,{n,0,99}]
LinearRecurrence[{4,-6,4,-1},{-11927,-9311,-7009,-5009},40] (* Harvey P. Dale, Jan 31 2017 *)

A218458(n):=2*n^3-163*n^2+2777*n-11927$
makelist(A218458(n),n,0,30); /* Martin Ettl, Nov 08 2012 */

G.f.: (-11927+38397*x-41327*x^2+14869*x^3)/(x-1)^4. - R. J. Mathar, Nov 07 2012

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Wesley Ivan Hurt, Apr 21 2021

cp's OEIS Frontend

A076809 a(n) = n^4 + 853n^3 + 2636n^2 + 3536n + 1753.

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

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A095946 Primes of the form 8n^4 - 2n^3 - 132n^2 + 318n + 80557.

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Magma

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A218456 2*n^3 - 313*n^2 + 6823*n - 13633.

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Mathematica

Maxima

Formula

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

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Mathematica

A218457 a(n) = 6*n^3 - 263*n^2 + 3469*n - 12841.

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Programs

Mathematica

PARI

Extensions

A095974 Evaluate n^4 - 93n^3 + 3196n^2 - 48008n + 265483 for n >= 0, record the primes.

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A096372 List of primes produced by a certain "prime-generating" quartic polynomial.

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Keywords

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Programs

A218456 2n^3 - 313n^2 + 6823*n - 13633.

A218457 a(n) = 6n^3 - 263n^2 + 3469*n - 12841.

A218458 a(n) = 2n^3 - 163n^2 + 2777*n - 11927.