A202018 -id:A202018 - cp's OEIS frontend

A331876 Number of primes of the form P(k) = k^2 + k + 41 for k <= 10^n, where P(k) is Euler's prime-generating polynomial A202018.

2, 11, 87, 582, 4149, 31985, 261081, 2208197, 19132653, 168806741, 1510676803

Offset: 0

Hugo Pfoertner, Jan 30 2020

			a(0) = 2 because 41 and 43 are the 2 primes generated for k <= 1 = 10^0.
a(1) = 11 because 41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151 are the 11 primes generated for k <= 10^1, (A202018(10) = 151).
a(3) = 87 because 87 terms of A202018(0..100) are prime. The 14 composites occur for k = A007634(1..14): 40, 41, 44, 49, 56, ...

Cf. A005846, A007634, A202018, A319906, A331877.

n=0;m=1;for(k=0,10^7,my(j=k^2+k+41);if(isprime(j),n++);if(k==m,m*=10;print1(n,", ")))

A330673 The possible v-factors for any A202018(k) (while A202018(k) = v * w, v and w are integers, w >= v >= 41, v = w iff w = 41, all such v-factors form the set V).

Original entry on oeis.org

41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 163, 167, 173, 179, 197, 199, 223, 227, 251, 263, 281, 307, 313, 347, 359, 367, 373, 379, 383, 397, 409, 419, 421, 439, 457, 461, 487, 499, 503, 523, 547, 563, 577, 593, 607, 641, 647, 653, 661, 673, 677, 691, 701, 709, 733

Offset: 0

Sergey Pavlov, Dec 23 2019

nonn

This is different from A257362: a(n) = A257362(n+1) for n=0..109, but a(110) = 1468 != 1471 = A257362(111). - Alois P. Heinz, Mar 02 2020
A kind of prime number sieve for the numbers of form x^2+x+41 (for so-called Euler primes, or A005846).
A set of all composite Euler numbers of form x^2+x+41 could be written as a 4-dimensional matrix m(i,j,t,u); a set of all terms of a(n) could be written as a 3-dimensional matrix v(i,j,t), since, for any integer u > -1, and for any w-factor that has the same values for i, j, t, we have the same v-factor (u = -1 iff w = 41); see formulas below.
Theorem. Let m be a term of A202018. Then m is composite iff m == 0 (mod v), where v is a term of a(n), v <= sqrt(m) (v = sqrt(m) iff m = 1681); otherwise, m is prime. Moreover, while m == 0 (mod p) (p is prime, p <= sqrt(m), p = sqrt(m) iff m = 1681), p is a term of a(n).
While i = 1, any v(i,t,j) is a term of both A202018 and a(n) (trivial).
Any w is a term of V and of a(n) which is the superset of V.

			Let  i = 3, t = 1, j = -1. Then v(i,t,j) = m(j) * i^2 + b + ja = 41 * 3^2 + 4 - 6 = 41 * 9 - 2 = 367, and 367 is a term of a(n).
We could find all terms of a(n) v < 10^n and then all Euler primes p < 10^(2n) (for n > 1, number of all numbers m such that are terms of A202018 (and any m < 10^(2n)) is 10^n; trivial).
Let 2n = 10; it's easy to establish that, while i > 49, any v(i,t,j)^2 > 10^10; thus, we can use 0 < i < 50 to find all numbers v < 10^5. While m is a term of A202018, m < 10^10, m is composite iff there is at least one v such that m == 0 (mod v); otherwise, m is prime. We could easily remove all "false" numbers v that cannot be divisors of any m. Let p' be a regular prime (p' is a term of A000040, but not of a(n)) such that any 3p' < UB(i); in our case, any 3p' < 50. Thus, we could try any v with p' = {2,3,5,7,11,13}; if v == 0 (mod p'), it is "false"; otherwise, there is at least one m < 10^10 such that m == 0 (mod v).

Sergey Pavlov, Table of n, a(n) for n = 0..312
Sergey Pavlov, The Euler primes sieve (for the primes of form x^2+x+41; draft paper)

Cf. A005846, A145292, A202018, A257362.

Let j = {-1;0;-2;1;-3;2;...;-(n+1);n}, m(-1) = 41, m(0) = 41, etc. (while j is negative, m(j) = A202018(-(j+1)); while j is nonnegative, m(j) = A202018(j)). Any term of a(n) could be written at least once as v(i,t+1,j) = m(j) * i^2 + b + ja, where i, t, and j are integers (j could be negative), i > 2; a = (i^2 - 2i) - 2i(t - 1), b = a - ((i^2 - 4)/4 - ((t - 1)^2 + 2(t - 1))), 0 < t < (i/2), while i is even; a = (i^2 - i) - 2i(t - 1), b = a - ((i^2 - 1)/4 - ((t - 1)^2 + (t - 1))), 0 < t < ((i + 1)/2), while i is odd (Note: v(i,1,j) = v(i,i/2,j), while i is even; v(i,1,j) = v(i,(i + 1)/2,j), while i is odd); at i = 2, v(2,1,j) = 4 * m(j) + 3 + 4j (at i = 2, we use only j < 0); at i = 1, v(1,1,j) = m(j) (at i = 1, we use only j >= 0; trivial).

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

Original entry on oeis.org

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

A221712 Hardy-Littlewood constant for x^2+x+41.

Original entry on oeis.org

3, 3, 1, 9, 7, 7, 3, 1, 7, 7, 4, 7, 1, 4, 2, 1, 6, 6, 5, 3, 2, 3, 5, 5, 6, 8, 5, 7, 6, 4, 9, 8, 8, 7, 9, 6, 6, 4, 6, 8, 5, 5, 4, 5, 8, 5, 6, 5, 2, 9, 8, 5, 8, 4, 9, 1, 5, 3, 9, 4, 0, 7, 2, 7, 9, 5, 0, 2, 6, 3, 3, 1, 0, 4, 2, 6, 1, 1, 8, 1, 4, 9, 7, 3, 7, 5, 5

Offset: 1

N. J. A. Sloane, Jan 26 2013

			3.31977317747142166532355685764988796646855...

Henri Cohen, Number Theory, Vol II: Analytic and Modern Tools, Springer (Graduate Texts in Mathematics 240), 2007.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 265-266.

Christian Axler and Mehdi Hassani, Carleman's inequality over prime numbers, Integers (2021) Vol. 21, #A53.
Karim Belabas and Henri Cohen, Computation of the Hardy-Littlewood constant for quadratic polynomials, PARI/GP script, 2020.
Lea Beneish and Christopher Keyes, How often does a cubic hypersurface have a rational point?, arXiv:2405.06584 [math.NT], 2024. See p. 23.
David Broadhurst, Five families of rapidly convergent evaluations of zeta values, arXiv:2401.08997 [math.NT], 2024.
Henri Cohen, High-precision calculation of Hardy-Littlewood constants, (1998).
Henri Cohen, High precision computation of Hardy-Littlewood constants. [Cached pdf version, with permission]
Stéphane Vinatier and William Reginald Alcorn, Mathematics as seen by an artist: Inspiring mathematical objects, Eur. Math. Soc. Mag. (2025) Vol. 135, 20-31. See p. 26.

Cf. A005846, A056561, A202018, A221713, A319906, A331876, A331877.

\\ See Belabas, Cohen link. Run as HardyLittlewood2(x^2+x+41)/2 after setting the required precision.

More terms from Hugo Pfoertner, Jan 31 2020

A056561 Numbers n such that n^2 + n + 41 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, 42, 43, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78

Offset: 1

Henry Bottomley, Jun 26 2000

Among first 100000 terms, the only run of 13 subsequent values >39 is 219..231. - Zak Seidov, Jan 28 2009
Number of terms less than 10^n: 1, 10, 86, 581, 4149, 31985, 261081, 2208197, 19132652, ... . - Robert G. Wilson v, Apr 20 2015
Complement of A007634. - Robert Israel, Apr 20 2015

			39 is in the sequence because 39^2+39+41=1601 which is prime but 40 is not because 40^2+40+41=1681=41*41.

P. Hoffman, Archimedes' Revenge, pp. 39-40,Penguin Books 1988.

Zak Seidov, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Euler Prime

Cf. A002837, A005846, A007634, A010051, A202018, A259645.

a056561 n = a056561_list !! (n-1)
a056561_list = filter ((== 1) . a010051' . a202018) [0..]
-- Reinhard Zumkeller, Jul 03 2015

[n: n in [0..80] |IsPrime(n^2 + n + 41)]; // Vincenzo Librandi, Sep 28 2012

select(t -> isprime(t^2+t+41), [$0..100]); # Robert Israel, Apr 20 2015

Select[Range[80], PrimeQ[#^2 + # + 41] &] (* Vincenzo Librandi, Sep 28 2012 *)

is(n)=isprime(n^2+n+41) \\ Charles R Greathouse IV, Sep 02 2015

a(n) = (sqrt(4*A005846(n)-163)-1)/2.

a(n) = A002837(n+1)-1. - Robert Price, Nov 08 2019

A145292 Composite numbers generated by the Euler polynomial x^2 + x + 41.

Original entry on oeis.org

1681, 1763, 2021, 2491, 3233, 4331, 5893, 6683, 6847, 7181, 7697, 8051, 8413, 9353, 10547, 10961, 12031, 13847, 14803, 15047, 15293, 16043, 16297, 17071, 18673, 19223, 19781, 20633, 21797, 24221, 25481, 26123, 26447, 26773, 27101, 29111

Offset: 1

Artur Jasinski, Oct 06 2008

nonn

The Euler polynomial x^2 + x + 41 gives primes for consecutive x from 0 to 39.
For numbers x for which x^2 + x + 41 is not prime see A007634.
Let P(x)=x^2 + x + 41. In view of identity P(x+P(x))=P(x)*P(x+1), all values of P(x+P(x)) are in the sequence. - Vladimir Shevelev, Jul 16 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A005846, A007634, A145293, A145294.

Intersection of A002808 and A202018; A010051.

a145292 n = a145292_list !! (n-1)
a145292_list = filter ((== 0) . a010051) a202018_list
-- Reinhard Zumkeller, Dec 09 2011

a = {}; Do[If[PrimeQ[x^2 + x + 41], null,AppendTo[a, x^2 + x + 41]], {x, 0, 500}]; a
Select[Table[x^2+x+41,{x,200}],CompositeQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 21 2018 *)

for(n=1,1e3,if(!isprime(t=n^2+n+41),print1(t", "))) \\ Charles R Greathouse IV, Dec 08 2011

a(n) ~ n^2. [Charles R Greathouse IV, Dec 08 2011]

A060566 a(n) = n^2 - 79*n + 1601.

Original entry on oeis.org

1601, 1523, 1447, 1373, 1301, 1231, 1163, 1097, 1033, 971, 911, 853, 797, 743, 691, 641, 593, 547, 503, 461, 421, 383, 347, 313, 281, 251, 223, 197, 173, 151, 131, 113, 97, 83, 71, 61, 53, 47, 43, 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, 1681

Offset: 0

Jason Earls, Apr 11 2001

a(n) is prime for 0 <= n <= 79. a(80) = 1681 = 41^2.
More than the usual number of terms are shown in order to display the initial 80 primes.
First 80 prime entries are palindromically distributed because a(n) = P(x) = x^2 + x + 41, with x = n - 40 and we observe that P(x) generates primes (A005846) for x = 0 through 39, along with the fact that P(-x) = P(x-1). - Lekraj Beedassy, Apr 24 2006

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 6.
C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Dover Publications, NY, 1966, p. 37, 147.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 115.

Andrew Howroyd, Table of n, a(n) for n = 0..1000 (terms 0..500 from Harry J. Smith and Miquel Cerda)
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002837, A005846, A202018.

[n^2-79*n+1601: n in [0..80]]; // Vincenzo Librandi, Feb 27 2017

A060566:=n->n^2-79*n+1601: seq(A060566(n), n=0..150); # Wesley Ivan Hurt, Jul 10 2017

Table[n^2-79*n+1601,{n,100}] (* or *) LinearRecurrence[{3,-3,1},{1523,1447,1373},100] (* Harvey P. Dale, Jan 14 2017 *)

a(n) = { n^2 - 79*n + 1601 } \\ Harry J. Smith, Jul 07 2009

From Vincenzo Librandi, Feb 27 2017: (Start)
G.f.: (1601 - 3280*x + 1681*x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
a(n) = (n-40)^2 + (n-40) + 41. - Miquel Cerda, Jul 10 2017
E.g.f.: exp(x)*(1601 - 78*x + x^2). - Elmo R. Oliveira, Feb 09 2025

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 16 2007

a(125) in b-file corrected by Andrew Howroyd, Feb 21 2018

A211773 Prime-generating polynomial: a(n) = 2n^2 - 108n + 1259.

Original entry on oeis.org

1259, 1153, 1051, 953, 859, 769, 683, 601, 523, 449, 379, 313, 251, 193, 139, 89, 43, 1, -37, -71, -101, -127, -149, -167, -181, -191, -197, -199, -197, -191, -181, -167, -149, -127, -101, -71, -37, 1, 43, 89, 139, 193, 251, 313, 379, 449, 523, 601, 683, 769, 859, 953

Offset: 0

Marius Coman, May 18 2012

This polynomial generates 92 primes (66 distinct ones) for 0 <= n <= 99 (in fact the next two terms are still primes but we keep the range 0-99, customary for comparisons), just three primes less than the record held by Euler's polynomial for n = m - 35, which is m^2 - 69*m + 1231 (see the link below), but having six distinct primes more than this one.
The nonprime terms in the first 100 are: 1 (taken twice), 1369 = 37^2, 1849 = 43^2, 4033 = 37*109, 5633 = 43*131, 7739 = 71*109 and 8251 = 37*223.
For n = 2*m - 34 we obtain the polynomial 8*m^2 - 488*m + 7243, which generates 31 primes in a row starting from m = 0 (polynomial already reported, see the link below).
For n = 4*m - 34 we obtain the polynomial 32*m^2 - 976*m + 7243, which generates 31 primes in row starting from m = 0.
The polynomial 2*n^2 + 40*n + 1, which generates the positive terms of this sequence in ascending order (i.e., a(37), ...), yields 10774009 distinct primes for 0 <= n < 49999999 while Euler's polynomial (n^2 - n + 41) gives 9967520 primes in same range. - Mikk Heidemaa, Feb 23 2016

Joe L. Mott and Kermite Rose, Prime-Producing Cubic Polynomials in Lecture Notes in Pure and Applied Mathematics (Vol. 220), Marcel Dekker Inc., 2001, pages 281-317.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Marius Coman, Ten prime-generating quadratic polynomials, Preprint 2015.
Joe L. Mott and Kermite Rose, Prime-Producing Cubic Polynomials.
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A060566, A202018.

[2*n^2-108*n+1259: n in [0..49]]; // Bruno Berselli, May 18 2012

Table[2 n^2 + 40 n + 1, {n, -37, 962}] (* Mikk Heidemaa, Feb 18 2016 *)

a(n)=2*n^2 - 108*n + 1259 \\ Charles R Greathouse IV, Jun 29 2017

G.f.: (1259 - 2624*x + 1369*x^2)/(1-x)^3. - Bruno Berselli, May 18 2012
a(n-37) = 2*n^2 + 40*n + 1. - Mikk Heidemaa, Feb 18 2016
From Elmo R. Oliveira, Feb 09 2025: (Start)
E.g.f.: exp(x)*(1259 - 106*x + 2*x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A238242 Primes p such that p^2+p+41 is also prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 97, 101, 103, 107, 113, 131, 137, 139, 149, 151, 157, 167, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 241, 257, 263, 269, 277, 281, 293, 307, 311, 313, 317, 337, 353

Offset: 1

K. D. Bajpai, Feb 20 2014

nonn

n^2 + n + 41 is Euler’s prime generating polynomial.

The first 12 terms in the sequence are the first 12 consecutive primes.

			13 is in the sequence because 13 is prime and 13^2+13+41 = 223 is also prime.
113 is in the sequence because 113 is prime and 113^2+113+41 =  12923 is also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..3372

Cf. A000040, A005846, A202018, A097823.

[p: p in PrimesUpTo(400)| IsPrime(p^2+p+41)]; // Vincenzo Librandi, Feb 22 2014

with(numtheory):KD := proc() local a,b; a:=ithprime(n); b:=a^2+a+41;  if isprime(b) then RETURN (a);  fi; end: seq(KD(), n=1..500);

Select[Prime[Range[200]],PrimeQ[#^2+#+41]&]

s=[]; forprime(p=2, 1000, if(isprime(p^2+p+41), s=concat(s, p))); s \\ Colin Barker, Feb 20 2014

A257362 Odd primes modulo which -163 is a square.

Original entry on oeis.org

41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 163, 167, 173, 179, 197, 199, 223, 227, 251, 263, 281, 307, 313, 347, 359, 367, 373, 379, 383, 397, 409, 419, 421, 439, 457, 461, 487, 499, 503, 523, 547, 563, 577, 593, 607, 641, 647, 653, 661, 673, 677, 691

Offset: 1

Robert Israel, Apr 20 2015

Contains A005846. The first members that are not in A005846 are 163 and 167.
Primes that divide some member of A202018.
Primes congruent to x^2 mod 163 for some x, 0 <= x <= 162.
Primes of the form x^2 + xy + 41y^2. Also, primes of the form x^2 - xy + 41y^2 with x and y nonnegative. - Jianing Song, Feb 19 2021

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A005846, A202018.

select(p -> isprime(p) and (p=163 or numtheory:-legendre(-163,p)=1), [seq(2*i+1,i=1..1000)]);
# Another Maple program is given in A296920. - N. J. A. Sloane, Dec 25 2017

Reap[For[p=3, p<1000, p = NextPrime[p], If[p==163 || KroneckerSymbol[-163, p] == 1, Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Apr 29 2019 *)

is(n)=isprime(n) && issquare(Mod(-163,n)) \\ Charles R Greathouse IV, Nov 28 2016

a(n) ~ 2n log n. - Charles R Greathouse IV, Nov 28 2016

cp's OEIS Frontend

A331876 Number of primes of the form P(k) = k^2 + k + 41 for k <= 10^n, where P(k) is Euler's prime-generating polynomial A202018.

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A330673 The possible v-factors for any A202018(k) (while A202018(k) = v * w, v and w are integers, w >= v >= 41, v = w iff w = 41, all such v-factors form the set V).

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

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GAP

Haskell

Magma

Maple

Mathematica

PARI

PARI

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Extensions

A221712 Hardy-Littlewood constant for x^2+x+41.

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PARI

Extensions

A056561 Numbers n such that n^2 + n + 41 is prime.

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Haskell

Magma

Maple

Mathematica

PARI

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A145292 Composite numbers generated by the Euler polynomial x^2 + x + 41.

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Haskell

Mathematica

PARI

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A060566 a(n) = n^2 - 79*n + 1601.

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A211773 Prime-generating polynomial: a(n) = 2n^2 - 108n + 1259.