A007635 -id:A007635 - cp's OEIS frontend

A005846 Primes of the form k^2 + k + 41.

41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, 1601, 1847, 1933, 2111, 2203, 2297, 2393, 2591, 2693, 2797

Offset: 1

N. J. A. Sloane

Note that 41 is the largest of Euler's Lucky numbers (A014556). - Lekraj Beedassy, Apr 22 2004
a(n) = A117530(13, n) for n <= 13: a(1) = A117530(13, 1) = A014556(6) = 41, A117531(13) = 13. - Reinhard Zumkeller, Mar 26 2006
The link to E. Wegrzynowski contains the following incorrect statement: "It is possible to find a polynomial of the form n^2 + n + B that gives prime numbers for n = 0, ..., A, A being any number." It is known that the maximum is A = 39 for B = 41. - Luis Rodriguez (luiroto(AT)yahoo.com), Jun 22 2008
Contrary to the last comment, Mollin's Theorem 2.1 shows that any A is possible if the Prime k-tuples Conjecture is assumed. - T. D. Noe, Aug 31 2009
a(n) can be generated by a recurrence based on the gcd in the type of Eric Rowland and Aldrich Stevens. See the recurrence in PARI under PROG. - Mike Winkler, Oct 02 2013
These primes are not prime in O_(Q(sqrt(-163))). Given p = n^2 + n + 41, we have ((2*n + 1)/2 - sqrt(-163)/2)*((2*n + 1)/2 + sqrt(-163)/2) = p, e.g., 1601 = 39^2 + 39 + 41 = (79/2 - sqrt(-163)/2)*(79/2 + sqrt(-163)/2). - Alonso del Arte, Nov 03 2017
From Peter Bala, Apr 15 2018: (Start)
The polynomial P(n) := n^2 + n + 41 takes distinct prime values for the 40 consecutive integers n = 0 to 39. It follows that the polynomial P(n-40) takes prime values for the 80 consecutive integers n = 0 to 79, consisting of the 40 primes above each taken twice. We note two consequences of this fact.
1) The polynomial P(2*n-40) = 4*n^2 - 158*n + 1601 also takes prime values for the 40 consecutive integers n = 0 to 39.
2) The polynomial P(3*n-40) = 9*n^2 - 237*n + 1601 takes prime values for the 27 consecutive integers n = 0 to 26 ( = floor(79/3)). In addition, calculation shows that P(3*n-40) also takes prime values for n from -13 to -1. Equivalently put, the polynomial P(3*n-79) = 9*n^2 - 471*n + 6203 takes prime values for the 40 consecutive integers n = 0 to 39. This result is due to Higgins. Cf. A007635 and A048059. (End)

			a(39) = 1601 = 39^2 + 39 + 41 is in the sequence because it is prime.
1681 = 40^2 + 40 + 41 is not in the sequence because 1681 = 41*41.

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 225.
R. K. Guy, Unsolved Problems Number Theory, Section A1.
O. Higgins, Another long string of primes, J. Rec. Math., 14 (1981/2) 185.
Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 137.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 139, 149.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 115.

Zak Seidov, Table of n, a(n) for n = 1..10000.
Phil Carmody, Drag Racing Prime Numbers! - _Vladimir Joseph Stephan Orlovsky_, Jul 24 2011
Richard K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
R. A. Mollin, Prime-producing quadratics, Amer. Math. Monthly 104 (1997), 529-544.
Jitender Singh, Prime numbers and factorization of polynomials, arXiv:2411.18366 [math.NT], 2024.
E. Wegrzynowski, Les formules simples qui donnent des nombres premiers en grande quantité
Eric Weisstein's World of Mathematics, Euler Prime
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

Cf. A048988, A007634, A056561, A002378, A007635.
Intersection of A000040 and A202018; A010051.
Cf. A048059.

Filtered(List([0..100],n->n^2+n+41),IsPrime); # Muniru A Asiru, Apr 22 2018

a005846 n = a005846_list !! (n-1)
a005846_list = filter ((== 1) . a010051) a202018_list
-- Reinhard Zumkeller, Dec 09 2011

[a: n in [0..55] | IsPrime(a) where a is n^2+n+ 41]; // Vincenzo Librandi, Apr 24 2018

for y from 0 to 10 do
U := y^2+y+41;
if isprime(U) = true then print(U) end if ;
end do:
# Matt C. Anderson, Jan 04 2013

Select[Table[n^2 + n + 41, {n, 0, 59}],PrimeQ] (* Alonso del Arte, Dec 08 2011 *)

for(n=1,1e3,if(isprime(k=n^2+n+41),print1(k", "))) \\ Charles R Greathouse IV, Jul 25 2011

{k=2; n=1; for(x=1, 100000, f=x^2+x+41; g=x^2+3*x+43; a=gcd(f, g-k); if(a>1, k=k+2); if(a==x+2-k/2, print(n" "a); n++))} \\ Mike Winkler, Oct 02 2013

a(n) = A056561(n)^2 + A056561(n) + 41.

More terms from Henry Bottomley, Jun 26 2000

A050267 Primes or negative values of primes in the sequence b(n) = 47n^2 - 1701n + 10181, n >= 0.

Original entry on oeis.org

10181, 8527, 6967, 5501, 4129, 2851, 1667, 577, -419, -1321, -2129, -2843, -3463, -3989, -4421, -4759, -5003, -5153, -5209, -5171, -5039, -4813, -4493, -4079, -3571, -2969, -2273, -1483, -599, 379, 1451, 2617, 3877, 5231, 6679, 8221, 9857, 11587, 13411, 15329, 17341, 19447, 21647, 31387

Offset: 1

Eric W. Weisstein

Terms are listed in the order of their appearance in sequence b.

This is a transformed version of the polynomial P(x) = 47*x^2 + 9*x - 5209 whose absolute value gives 43 distinct primes for -24 <= x <= 18, found by G. W. Fung in 1988. - Hugo Pfoertner, Dec 13 2019

R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004 (ISBN 0-387-20860-7); see Section A17, p. 59.
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 = 1..1000
G. W. Fung and H. C. Williams, Quadratic polynomials which have a high density of prime values, Math. Comput. 55(191) (1990), 345-353.
Carlos Rivera, Problem 12: Prime producing polynomials, The Prime Puzzles & Problems Connection.
Jitender Singh, Prime numbers and factorization of polynomials, arXiv:2411.18366 [math.NT], 2024.
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.

Cf. A002383, A005471, A005846, A007635, A022464, A027753, A027755, A027758, A048059, A050267, A050268, A116206, A117081, A267252.

lst={};Do[p=47*n^2-1701*n+10181;If[PrimeQ[p],AppendTo[lst,p]],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 29 2009 *)
Select[Table[47n^2-1701n+10181,{n,0,50}],PrimeQ] (* Harvey P. Dale, Oct 03 2011 *)

[n | n <- apply(m->47*m^2-1701*m+10181, [0..100]), isprime(abs(n))] \\ Charles R Greathouse IV, Jun 18 2017

Edited by N. J. A. Sloane, May 10 2007
Further edited by Klaus Brockhaus, Mar 20 2010
More terms (to distinguish from quadratic) from Charles R Greathouse IV, Jun 18 2017

A271980 Numbers k such that 3k^2 + 39k + 37 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, 19, 20, 21, 22, 23, 25, 26, 29, 30, 31, 32, 33, 34, 35, 38, 39, 40, 41, 42, 44, 45, 46, 48, 49, 51, 52, 53, 54, 55, 57, 58, 59, 60, 63, 64, 66, 68, 69, 70, 71, 72, 79, 84, 86, 88, 89, 90, 91, 92

Offset: 1

Robert Price, Apr 17 2016

From Peter Bala, Apr 16 2018: (Start)
Let P(n) = 3*n^2 + 39*n + 37. The absolute values of the polynomial P(2*n - 29) = 12*n^2 - 270*n + 1429 for n from 0 to 27 are distinct primes, except at n = 14 when the value is 1.
The absolute values of the polynomial 3*P((n - 20)/3) = n^2 - n - 269 for n from 0 to 42 are either prime or 3 times a prime.
The absolute values of the polynomial 3*P((4*n - 89)/3) = 16*n^2 - 556*n + 4561 for n from 0 to 27 are either prime or 3 times a prime. (End)

			4 is in this sequence since 3*4^2 + 39*4 + 37 = 48+156+37 = 241 is prime.

Robert Price, Table of n, a(n) for n = 1..3510
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A256585, A050265 - A050268, A005846, A007641, A007635, A048988.

[n: n in [0..100] |IsPrime(3*n^2+39*n+37)]; // Vincenzo Librandi, Apr 19 2018

Select[Range[0, 100], PrimeQ[3*#^2 + 39*# + 37] &]

isok(n) = isprime(3*n^2 + 39*n + 37); \\ Michel Marcus, Apr 17 2016

lista(nn) = for(n=0, nn, if(ispseudoprime(3*n^2+39*n+37), print1(n, ", "))); \\ Altug Alkan, Apr 18 2016

A272074 Numbers k such that k^4 + 29*k^2 + 101 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, 21, 23, 26, 31, 32, 34, 35, 37, 43, 44, 45, 47, 49, 53, 56, 60, 61, 62, 66, 67, 68, 70, 71, 72, 74, 75, 79, 80, 81, 84, 85, 89, 90, 91, 93, 96, 99

Offset: 1

Robert Price, Apr 19 2016

			4 is in this sequence since 4^4 + 29*4^2 + 101 = 256+464+101 = 821 is prime.

Robert Price, Table of n, a(n) for n = 1..2264
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A050265 - A050268, A005846, A007641, A007635, A048988, A256585.

Cf. A271980, A272075.

Select[Range[0,100],PrimeQ[#^4+29#^2+101]&] (* Harvey P. Dale, Dec 15 2020 *)

lista(nn) = for(n=0, nn, if(ispseudoprime(n^4+29*n^2+101), print1(n, ", "))); \\ Altug Alkan, Apr 19 2016

A272075 Primes of the form k^4 + 29*k^2 + 101.

Original entry on oeis.org

101, 131, 233, 443, 821, 1451, 2441, 3923, 6053, 9011, 13001, 18251, 25013, 33563, 44201, 57251, 73061, 92003, 114473, 140891, 207371, 295283, 476681, 951491, 1078373, 1369961, 1536251, 1913963, 3472523, 3804341, 4159451, 4943843, 5834531, 7972043, 9925541

Offset: 1

Robert Price, Apr 19 2016

			233 is prime and it is in this sequence since 233 = 2^4 + 29*2^2 + 101.

Robert Price, Table of n, a(n) for n = 1..2264
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

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

Cf. A271980, A272074.

n = Range[0, 100]; Select[#^4 + 29#^2 + 101, PrimeQ[#] &]

lista(nn) = for(n=0, nn, if(ispseudoprime(p=n^4+29*n^2+101), print1(p, ", "))); \\ Altug Alkan, Apr 19 2016

A048059 Primes of the form k^2 + k + 11.

Original entry on oeis.org

11, 13, 17, 23, 31, 41, 53, 67, 83, 101, 167, 193, 251, 283, 317, 353, 431, 563, 661, 823, 881, 941, 1201, 1493, 1571, 1733, 2081, 2267, 2663, 2767, 3203, 3433, 3671, 3793, 3917, 4567, 4703, 5413, 5711, 6173, 6491, 6653, 6983, 7151, 7321, 8753, 8941, 9323, 10111

Offset: 1

N. J. A. Sloane

From Peter Bala, Apr 15 2018: (Start)
The polynomial P(n) := n^2 + n + 11 takes distinct prime values for the 10 consecutive integers n = 0 to 9. It follows that the polynomial P(n-10) = (n - 10)^2 + (n - 10) + 11 takes prime values for the 20 consecutive integers n = 0 to 19, consisting of the 10 primes above each taken twice. We note two consequences of this fact.
1) The polynomial P(2*n-10) = 4*n^2 - 38*n + 101 also takes prime values for the 10 consecutive integers n = 0 to 9.
2)The polynomial P(3*n-10) = 9*n^2 - 57*n + 101 takes prime values for the 7 consecutive integers n = 0 to 6 (= floor(19/3)). In addition, calculation shows that P(3*n-10) also takes prime values for n from -3 to -1. Equivalently put, the polynomial P(3*n-19) = 9*n^2 - 111*n + 353 takes prime values for the 10 consecutive integers n = 0 to 9. Cf. A007635 and A005846. (End)

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

Cf. A005846, A007635, A048058, A048097, A160548.

[ a: n in [0..200] | IsPrime(a) where a is n^2+n+11 ]; // Vincenzo Librandi, Dec 07 2011

lst={}; Do[p=n^2+n+11; If[PrimeQ[p], AppendTo[lst,p]], {n,0,5*5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 27 2009 *)
Select[Table[n^2+n+11, {n,0,600}], PrimeQ] (* Vincenzo Librandi, Dec 07 2011 *)

a(n) = A048058(A048097(n)). - Elmo R. Oliveira, Apr 20 2025

A272160 Primes of the form abs(8n^2 - 488n + 7243) in order of increasing nonnegative values of n.

Original entry on oeis.org

7243, 6763, 6299, 5851, 5419, 5003, 4603, 4219, 3851, 3499, 3163, 2843, 2539, 2251, 1979, 1723, 1483, 1259, 1051, 859, 683, 523, 379, 251, 139, 43, 37, 101, 149, 181, 197, 197, 181, 149, 101, 37, 43, 139, 251, 379, 523, 683, 859, 1051, 1259, 1483, 1723, 1979

Offset: 1

Robert Price, Apr 21 2016

nonn

			5419 is in this sequence since 8*4^2 - 488*4 + 7243 = 128-1952+7243 = 5419 is prime.

Robert Price, Table of n, a(n) for n = 1..4058
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

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

Cf. A271980, A272074, A272075, A272118, A272159.

n = Range[0, 100]; Select[Abs[8n^2 - 488n + 7243], PrimeQ[#] &]

lista(nn) = for(n=0, nn, if(isprime(p=abs(8*n^2-488*n+7243)), print1(p, ", "))); \\ Altug Alkan, Apr 21 2016

A271144 Primes of the form 42k^3 + 270k^2 - 26436*k + 250703 in order of increasing k.

Original entry on oeis.org

250703, 224579, 199247, 174959, 151967, 130523, 110879, 93287, 77999, 65267, 55343, 48479, 44927, 44939, 48767, 56663, 68879, 85667, 107279, 133967, 165983, 203579, 247007, 296519, 352367, 414803, 484079, 560447, 644159, 735467, 834623, 941879, 1057487

Offset: 1

Robert Price, Apr 23 2016

nonn

			151967 is prime and it is in this sequence since 151967 = 42*4^3 + 270*4^2 - 26436*4 + 250703.

Robert Price, Table of n, a(n) for n = 1..3092
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A050265 - A050268, A005846, A007641, A007635, A048988, A256585

Cf. A271980, A272074, A272075, A272118, A272159, A271143 (associated k).

n = Range[0, 100]; Select[42n^3 + 270n^2 - 26436n + 250703, PrimeQ[#] &]

A272159 Numbers k such that abs(8k^2 - 488k + 7243) 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, 57, 58, 59, 60, 61, 64, 65, 66, 67, 71

Offset: 1

Robert Price, Apr 21 2016

nonn

From Robert Israel, Apr 21 2016: (Start)
n such that either n <= 61 or 8n^2 - 488n + 7243 is prime.
The first number not in the sequence is 62. (End)

			4 is in this sequence since 8*4^2 - 488*4 + 7243 = 128-1952+7243 = 5419 is prime.

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

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

Cf. A271980, A272074, A272075, A272160.

select(n -> isprime(abs(8*n^2 - 488*n + 7243)), [$0..1000]); # Robert Israel, Apr 21 2016

Select[Range[0, 100], PrimeQ[8#^2 - 488# + 7243] &]

lista(nn) = for(n=0, nn, if(isprime(abs(8*n^2-488*n+7243)), print1(n, ", "))); \\ Altug Alkan, Apr 21 2016

A271143 Numbers k such that 42k^3 + 270k^2 - 26436*k + 250703 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, 41, 43, 44, 48, 51, 54, 55, 56, 58, 61, 62, 63, 64, 65, 66, 67, 69, 71, 76, 78, 79, 84, 87, 88, 89, 90, 92

Offset: 1

Robert Price, Apr 23 2016

nonn

40 is the first value not in the sequence.

			4 is in this sequence since 42*4^3 + 270*4^2 - 26436*4 + 250703 = 151967, which is prime.

Robert Price, Table of n, a(n) for n = 1..3092
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A050265-A050268, A005846, A007641, A007635, A048988, A256585, A271980, A272074, A272075, A272160, A271144.

Select[Range[0, 100], PrimeQ[42#^3 + 270#^2 - 26436# + 250703] &]

is(n)=isprime(42*n^3+270*n^2-26436*n+250703) \\ Charles R Greathouse IV, Feb 17 2017

cp's OEIS Frontend

A005846 Primes of the form k^2 + k + 41.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Haskell

Magma

Maple

Mathematica

PARI

PARI

Formula

Extensions

A050267 Primes or negative values of primes in the sequence b(n) = 47*n^2 - 1701*n + 10181, n >= 0.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A271980 Numbers k such that 3*k^2 + 39*k + 37 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

A272074 Numbers k such that k^4 + 29*k^2 + 101 is prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A272075 Primes of the form k^4 + 29*k^2 + 101.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A048059 Primes of the form k^2 + k + 11.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A272160 Primes of the form abs(8n^2 - 488n + 7243) in order of increasing nonnegative values of n.

Views

Author

Keywords

A050267 Primes or negative values of primes in the sequence b(n) = 47n^2 - 1701n + 10181, n >= 0.

A271980 Numbers k such that 3k^2 + 39k + 37 is prime.

A271144 Primes of the form 42k^3 + 270k^2 - 26436*k + 250703 in order of increasing k.

A272159 Numbers k such that abs(8k^2 - 488k + 7243) is prime.

A271143 Numbers k such that 42k^3 + 270k^2 - 26436*k + 250703 is prime.