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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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)

Examples

			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.
		

References

• 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.

Links

• 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

Crossrefs

Cf. A048988, A007634, A056561, A002378, A007635.

Intersection of A000040 and A202018; A010051.

Cf. A048059.

Programs

GAP

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

Haskell

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

Magma

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

Maple

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

Mathematica

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

PARI

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

PARI

{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

Formula

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

Extensions

More terms from Henry Bottomley, Jun 26 2000