A050267 -id:A050267 - cp's OEIS frontend

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

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

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

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

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[#] &]

A217439 Primes or negative values of primes of the form 8n^2 - 298n + 2113 for n >= 0.

Original entry on oeis.org

2113, 1823, 1549, 1291, 1049, 823, 613, 419, 241, 79, -67, -197, -311, -409, -491, -557, -607, -641, -659, -661, -647, -617, -571, -509, -431, -337, -227, -101, 41, 199, 373, 563, 769, 991, 1229, 1483, 1753, 2039, 2341, 2659, 3343, 3709, 4091, 4903, 5333, 5779, 6719, 7213

Offset: 1

Pedja Terzic, Oct 03 2012

Terms are listed in the order of appearance. The absolute values are prime for 0 <= n <= 39.

Eric Weisstein's World of Mathematics, Prime Generating Polynomials

Cf. A050267, A050268, A217440.

Select[Table[8*n^2 - 298*n + 2113, {n, 0, 50}], PrimeQ[#]&]

[n | n <- apply(m->8*m^2-298*m+2113, [0..100]), isprime(abs(n))] \\ Charles R Greathouse IV, Jun 18 2017

More terms (to distinguish from quadratic) from Charles R Greathouse IV, Jun 18 2017

A217440 Primes or negative values of primes of the form 8n^2 - 326n + 2659 for n >= 0.

Original entry on oeis.org

2659, 2341, 2039, 1753, 1483, 1229, 991, 769, 563, 373, 199, 41, -101, -227, -337, -431, -509, -571, -617, -647, -661, -659, -641, -607, -557, -491, -409, -311, -197, -67, 79, 241, 419, 613, 823, 1049, 1291, 1549, 1823, 2113, 2741, 3079, 3433, 3803

Offset: 1

Pedja Terzic, Oct 03 2012

Terms are listed in the order of appearance. The absolute values are primes for 0 <= n <= 39.

Eric Weisstein's World of Mathematics, Prime Generating Polynomials

Cf. A050267, A050268, A217439.

Select[Table[8*n^2 - 326*n + 2659, {n, 0, 50}], PrimeQ[#]&]

[n | n <- apply(m->8*m^2-326*m+2659, [0..100]), isprime(abs(n))] \\ Charles R Greathouse IV, Jun 18 2017

More terms (to distinguish from quadratic) from Charles R Greathouse IV, Jun 18 2017

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A268200 Nonnegative numbers n such that abs(n^4 - 97n^3 + 3294n^2 - 45458n + 213589) is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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.

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

A217439 Primes or negative values of primes of the form 8n^2 - 298n + 2113 for n >= 0.

A217440 Primes or negative values of primes of the form 8n^2 - 326n + 2659 for n >= 0.