A048059 -id:A048059 - 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

A007635 Primes of form n^2 + n + 17.

Original entry on oeis.org

17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257, 359, 397, 479, 523, 569, 617, 719, 773, 829, 887, 947, 1009, 1277, 1423, 1499, 1657, 1823, 1997, 2087, 2179, 2273, 2467, 2879, 3209, 3323, 3557, 3677, 3923, 4049, 4177, 4987, 5273

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

a(n) = A117530(7,n) for n <= 7: a(1) = A117530(7,1) = A014556(5) = 17, A117531(7) = 7. - Reinhard Zumkeller, Mar 26 2006
Note that the gaps between terms increases by 2*k from k = 1 to 15: 19 - 17 = 2, 23 - 19 = 4, 29 - 23 = 6 and so on until 257 - 227 = 30 then fails at 289 - 257 = 32 since 289 = 17^2. - J. M. Bergot, Mar 18 2017
From Peter Bala, Apr 15 2018: (Start)
The polynomial P(n):= n^2 + n + 17 takes distinct prime values for the 16 consecutive integers n = 0 to 15. It follows that the polynomial P(n - 16) takes prime values for the 32 consecutive integers n = 0 to 31, consisting of the 16 primes above each taken twice. We note two consequences of this fact.
1) The polynomial P(2*n - 16) = 4*n^2 - 62*n + 257 also takes prime values for the 16 consecutive integers n = 0 to 15.
2)The polynomial P(3*n - 16) = 9*n^2 - 93*n + 257 takes prime values for the 11 consecutive integers n = 0 to 10 ( = floor(31/3)). In addition, calculation shows that P(3*n-16) also takes prime values for n from -5 to -1. Equivalently put, the polynomial P(3*n-31) = 9*n^2 - 183*n + 947 takes prime values for the 16 consecutive integers n = 0 to 15. Cf. A005846 and A048059. (End)
The primes in this sequence are not primes in the ring of integers of Q(sqrt(-67)). If p = n^2 + n + 17, then ((2n + 1)/2 - sqrt(-67)/2)((2n + 1)/2 + sqrt(-67)/2) = p. For example, 3^2 + 3 + 17 = 29 and (7/2 - sqrt(-67)/2)(7/2 + sqrt(-67)/2) = 29 also. - Alonso del Arte, Nov 27 2019

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.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 96.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime-generating Polynomial.

Cf. A005846, A028823, A048059, A160548.

[a: n in [0..250]|IsPrime(a) where a is n^2+n+17]; // Vincenzo Librandi, Dec 23 2010

Select[Table[n^2 + n + 17, {n, 0, 99}], PrimeQ] (* Alonso del Arte, Nov 27 2019 *)

select(isprime, vector(100,n,n^2+n+17)) \\ Charles R Greathouse IV, Jul 12 2016

from sympy import isprime
it = (n**2 + n + 17 for n in range(250))
print([p for p in it if isprime(p)]) # Indranil Ghosh, Mar 18 2017

a(n) = A028823(n)^2 + A028823(n) + 17. - Seiichi Manyama, Mar 19 2017

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

A048058 a(n) = n^2 + n + 11.

Original entry on oeis.org

11, 13, 17, 23, 31, 41, 53, 67, 83, 101, 121, 143, 167, 193, 221, 251, 283, 317, 353, 391, 431, 473, 517, 563, 611, 661, 713, 767, 823, 881, 941, 1003, 1067, 1133, 1201, 1271, 1343, 1417, 1493, 1571, 1651, 1733, 1817, 1903, 1991, 2081, 2173, 2267, 2363, 2461, 2561

Offset: 0

N. J. A. Sloane

Fontebasso lists this as a prime-generating polynomial due to Legendre, but doesn't give a reference. - Charles R Greathouse IV, Jun 30 2021

Seiichi Manyama, Table of n, a(n) for n = 0..10000
P. A. Fontebasso, Un'altra formula che dà una serie limitata di numeri primi, Supplemento al Periodico di matematica (1899), p. 130.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002522, A048059, A048097, A176271.

with(combinat):seq(fibonacci(3, n)+n+10, n=0..45); # Zerinvary Lajos, Jun 07 2008

f[n_]:=n^2+n+11;f[Range[0,100]] (* Vladimir Joseph Stephan Orlovsky, Mar 12 2011*)

a(n)=n^2+n+11 \\ Charles R Greathouse IV, Jun 29 2015

For n > 4: a(n) = A176271(n+1,6). - Reinhard Zumkeller, Apr 13 2010
a(n) = 2*n + a(n-1) (with a(0)=11). - Vincenzo Librandi, Aug 06 2010
Sum_{n>=0} 1/a(n) = Pi*tanh(Pi*sqrt(43)/2)/sqrt(43). - Amiram Eldar, Jan 17 2021
From Elmo R. Oliveira, Oct 28 2024: (Start)
G.f.: (11 - 20*x + 11*x^2)/(1 - x)^3.
E.g.f.: (11 + 2*x + x^2)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A048097 Numbers k such that k^2 + k + 11 is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 15, 16, 17, 18, 20, 23, 25, 28, 29, 30, 34, 38, 39, 41, 45, 47, 51, 52, 56, 58, 60, 61, 62, 67, 68, 73, 75, 78, 80, 81, 83, 84, 85, 93, 94, 96, 100, 101, 103, 106, 108, 113, 114, 122, 123, 124, 125, 127, 130, 135

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A048058, A048059.

[n: n in [0..150] |IsPrime(n^2+n+11)]; // Vincenzo Librandi, Aug 06 2010

Select[Range[0,150],PrimeQ[#^2+#+11]&] (* Harvey P. Dale, May 18 2011 *)

is(n)=isprime(n^2+n+11) \\ Charles R Greathouse IV, Feb 17 2017

A117530 Triangle read by rows: T(n,k) = k^2 - k + prime(n), 1<=k<=n.

Original entry on oeis.org

2, 3, 5, 5, 7, 11, 7, 9, 13, 19, 11, 13, 17, 23, 31, 13, 15, 19, 25, 33, 43, 17, 19, 23, 29, 37, 47, 59, 19, 21, 25, 31, 39, 49, 61, 75, 23, 25, 29, 35, 43, 53, 65, 79, 95, 29, 31, 35, 41, 49, 59, 71, 85, 101, 119, 31, 33, 37, 43, 51, 61, 73, 87, 103, 121, 141, 37, 39, 43, 49, 57

Offset: 1

Reinhard Zumkeller, Mar 25 2006

A117531 gives the number of primes in the n-th row;

if T(n,1) is a Lucky Number of Euler then A117531(n)=n, see A014556.

			T(5,k)=A048058(k)=A048059(k), 1<=k<=5: T(5,1)=A014556(4)=11;
T(7,k)=A007635(k), 1<=k<=7: T(7,1)=A014556(5)=17;
T(13,k)=A005846(k), 1<=k<=13: T(13,1)=A014556(6)=41.

Robert Price, Table of n, a(n) for n = 1..10011
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial
Eric Weisstein's World of Mathematics, Lucky Number of Euler

Cf. A005846, A007635, A014556, A048058, A048059, A117531.

Table[k^2 - k + Prime[n], {n, 141}, {k, n}] // Flatten (* Robert Price, Apr 19 2025 *)

T(n,k) = k^2 - k + prime(n) \\ Charles R Greathouse IV, Apr 24 2015

T(n,1) = A000040(k).
T(n,2) = A052147(k) for k>1.
For 1

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

A302445 Triangle read by rows: row n gives primes of form k^2 + n - k for 0 < k < n.

Original entry on oeis.org

2, 3, 5, 5, 7, 11, 17, 7, 13, 19, 37, 11, 29, 11, 13, 17, 23, 31, 41, 53, 67, 83, 101, 13, 19, 43, 103, 17, 71, 197, 17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257, 19, 31, 61, 109, 151, 229, 23, 41, 131, 293, 401, 23, 29, 43, 53, 79, 113, 179, 233, 263, 443

Offset: 2

Seiichi Manyama, Apr 08 2018

			  n\k|  1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16
  ---+-----------------------------------------------------------------------
    2|  2;
    3|  3,  5;
    4|
    5|  5,  7, 11, 17;
    6|
    7|  7,   , 13, 19,   , 37;
    8|
    9|   , 11,   ,   , 29,   ,   ,   ;
   10|
   11| 11, 13, 17, 23, 31, 41, 53, 67, 83, 101;
   12|
   13| 13,   , 19,   ,   , 43,   ,   ,   , 103,    ,    ;
   14|
   15|   , 17,   ,   ,   ,   ,   , 71,   ,    ,    ,    ,    , 197;
   16|
   17| 17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257;

Seiichi Manyama, Rows n = 2..421, flattened
Eric Weisstein's World of Mathematics, Lucky Number of Euler
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial
Wikipedia, Lucky numbers of Euler

Row n: A027753 (n=3), A027755 (n=5), A048059 (n=11), A007635 (n=17), A005846 (n=41).

Cf. A000040, A014556, A302826.

a:=Filtered(Flat(List([1..10],n->List([1..n],k->k^2+n-k))),IsPrime); # Muniru A Asiru, Apr 09 2018

Map[Union@ Select[#, PrimeQ] &, Table[k^2 + n - k, {n, 23}, {k, 0, n}]] // Flatten (* Michael De Vlieger, Apr 10 2018 *)

A128878 Primes of the form 47n^2 - 1701n + 10181.

Original entry on oeis.org

10181, 8527, 6967, 5501, 4129, 2851, 1667, 577, 379, 1451, 2617, 3877, 5231, 6679, 8221, 9857, 11587, 13411, 15329, 17341, 19447, 21647, 31387, 34057, 36821, 39679, 45677, 48817, 52051, 65927, 81307, 89561, 102647, 107197, 116579, 126337, 131357

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Apr 17 2007

nonn

Primes are given in the order in which they arise for increasing n.
Polynomial generates 22 primes for 0 <= n <= 42, i.e., for n = 0, 1, 2, 3, 4, 5, 6, 7, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42.
If the definition is replaced by "Numbers n of the form 47*k^2 - 1701*k + 10181 such that either n or -n is a prime" we get (essentially) A050267.

			47k^2 - 1701k + 10181 = 21647 for k = 42.

R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, ISBN 0-387-20860-7, Section A17, page 59.

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 and Problems Connection.

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

Select[Table[47*n^2 - 1701*n + 10181, {n, 0, 100}], # > 0 && PrimeQ[#] &] (* T. D. Noe, Aug 02 2011 *)

Edited by Klaus Brockhaus, Apr 22 2007 and by N. J. A. Sloane, May 05 2007 and May 06 2007

Showing 1-10 of 16 results. Next

cp's OEIS Frontend

A175154 a(n) = prime index of A048059(n).

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

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A007635 Primes of form n^2 + n + 17.

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

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A048058 a(n) = n^2 + n + 11.

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A048097 Numbers k such that k^2 + k + 11 is prime.

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A117530 Triangle read by rows: T(n,k) = k^2 - k + prime(n), 1<=k<=n.

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

A128878 Primes of the form 47n^2 - 1701n + 10181.