A056561 -id:A056561 - 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

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

A202018 a(n) = n^2 + n + 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, 1681, 1763, 1847, 1933, 2021, 2111, 2203, 2297, 2393

Offset: 0

Reinhard Zumkeller, Dec 08 2011

Euler's famous prime-generating polynomial; a(0) through a(39) are all prime.

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 225.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 138-139, 145.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A060566, A010051, A000040, A002808.
Cf. A002378, A005846, A145292.
Cf. A056561.

List([0..50], n -> n^2 +n+41); # G. C. Greubel, Dec 04 2018

Haskell
```
a202018 = (+ 41) . a002378
```

[n^2 + n + 41 : n in [0..50]]; // Wesley Ivan Hurt, Sep 28 2014

A202018:=n->n^2+n+41: seq(A202018(n), n=0..50); # Wesley Ivan Hurt, Sep 28 2014

Table[n^2 + n + 41, {n, 0, 49}] (* Alonso del Arte, Dec 08 2011 *)

a(n)=n^2+n+41 \\ Charles R Greathouse IV, Dec 08 2011

[n^2+n+41 for n in range(50)] # G. C. Greubel, Dec 04 2018

(0 to 49).map((n: Int) => n * n + n + 41) // Alonso del Arte, Nov 29 2018

a(n) = A005846(n) for n < 41, a(41) = A145292(1);
Union of A005846 (primes) and A145292 (composites);
a(n) = A002378(n) + 41.
a(a(n) + n) = a(n)*a(n+1). - Vladimir Shevelev, Jul 16 2012 (This identity holds for all sequences of the form n^2 + n + c, Joerg Arndt, Jul 17 2012).
a(0) = 41 and for n > 0, a(n) = a(n-1) + 2*n. - Jean-Christophe Hervé, Sep 27 2014
From Colin Barker, Sep 28 2014: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: (41*x^2 - 80*x + 41) / (1-x)^3. (End)
a(n) = 2*a(n-1) - a(n-2) + 2. - Vincenzo Librandi, Mar 04 2016
E.g.f.: (x^2 + 2*x + 41)*exp(x). - Robert Israel, Mar 10 2016
Sum_{n>=0} 1/a(n) = tanh(sqrt(163)*Pi/2)*Pi/sqrt(163). - Amiram Eldar, May 12 2025

A002837 Numbers k such that k^2 - k + 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, 40, 43, 44, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72

Offset: 1

N. J. A. Sloane

Leonhard Euler published this prime-generating formula in 1772. - Harvey P. Dale, Sep 23 2020

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 6.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A007634, A056561, A060566.

[n: n in [0..100] |IsPrime(n^2-n+41)]; // Vincenzo Librandi, Nov 21 2010

A002837:=n->`if`(isprime(n^2-n+41),n,NULL): seq(A002837(n), n=0..100); # Wesley Ivan Hurt, Oct 21 2014

Select[Range[0,100],PrimeQ[#^2-#+41]&] (* Harvey P. Dale, May 27 2012 *)

v=[ ]; for(n=0,100, if(isprime(n^2-n+41),v=concat(v,n),)); v

a(n) = A056561(n-1) + 1, n > 1. - Robert Price, Nov 08 2019

A090102 Leading prime in each set of 7 arising in A090101.

Original entry on oeis.org

11, 516811, 20402952601, 196260616589761, 239536538008051, 426813020692661, 2681027962124411, 3605832801512401, 6450361508166761, 10392841156929031, 13162202092936411, 13655671002023851, 14501847401205811

Offset: 1

Labos Elemer, Dec 15 2003

nonn

			a[15] = 69981018761651281 is first of following chain: {69981018761651281, 69981019944706811, 69981021127762351, 69981022310817901, 69981023493873461, 69981024676929031, 69981025859984611} = {P[k], P[k+1], ..., P[k+6]}, where k = A090101[15] and P[x] = 5x^2+5x+1. See A090562, A090563.

Cf. A090562, A090563, A090100, A090101, A056561.

po[x_] := 5*x^2+5*x+1 Do[s=po[n];s0=po[n];s1=po[n+1];s2=po[n+2];s3=po[n+3];s4=po[n+4]; s5=po[n+5];s6=po[n+6];If[IntegerQ[n/100000], Print[{n}]]; If[PrimeQ[s0]&&PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s3]&&PrimeQ[s4]&&PrimeQ[s5] &&PrimeQ[s6], Print[s0]], {n, 1, 120000000}]

A259645 Numbers m such that m^2 + 1, 3*m - 1 and m^2 + m + 41 are all prime.

Original entry on oeis.org

1, 2, 4, 6, 10, 14, 16, 20, 24, 36, 66, 90, 94, 116, 120, 134, 150, 156, 160, 206, 240, 280, 340, 350, 384, 396, 430, 436, 470, 536, 634, 690, 700, 714, 780, 826, 864, 930, 946, 960, 1004, 1124, 1150, 1176, 1294, 1316, 1376, 1410, 1430, 1494, 1644, 1674

Offset: 1

Reinhard Zumkeller, Jul 03 2015

nonn

This sequence is infinite if the generalized Dickson's conjecture holds.

			.            | (i, j, k) such that |        corresponding
.            | a(n) = A005574(i)   |        prime triples
.     |      |      = A087370(j)   |        let m = a(n):
.   n | a(n) |      = A056561(k)   |  (m^2+1, 3*m-1, m^2+m+41)
.  ---+------+---------------------+--------------------------
.   1 |    1 |     (1,  1,  2)     |        (2,   2,   43)
.   2 |    2 |     (2,  2,  3)     |        (5,   5,   47)
.   3 |    4 |     (3,  3,  5)     |       (17,  11,   61)
.   4 |    6 |     (4,  4,  7)     |       (37,  17,   83)
.   5 |   10 |     (5,  6, 11)     |      (101,  29,  151)
.   6 |   14 |     (6,  7, 13)     |      (197,  41,  251)
.   7 |   16 |     (7,  8, 15)     |      (257,  47,  313)
.   8 |   20 |     (8, 10, 21)     |      (401,  59,  461)
.   9 |   24 |     (9, 11, 25)     |      (597,  71,  641)
.  10 |   36 |    (11, 15, 37)     |     (1297, 107, 1373)
.  11 |   66 |    (15, 24, 61)     |     (4357, 197, 4463)
.  12 |   90 |    (18, 31, 79)     |     (8101, 269, 8231)  .

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Wikipedia, Bunyakovsky conjecture
Wikipedia, Dickson's conjecture

Intersection of A005574, A087370 and A056561.

import Data.List.Ordered (isect)
a259645 n = a259645_list !! (n-1)
a259645_list = a005574_list `isect` a087370_list `isect` a056561_list

Select[Range[100], AllTrue[{#^2 + 1, 3 # - 1, #^2 + # + 41}, PrimeQ] &] (* Robert Price, Apr 19 2025 *)

A188424 Number of primes of the form k^2 + k + 2n - 1 for k = 0..2n-1.

Original entry on oeis.org

1, 2, 4, 4, 2, 10, 4, 3, 16, 6, 5, 10, 10, 5, 13, 14, 3, 10, 16, 7, 40, 8, 6, 26, 12, 9, 19, 14, 9, 34, 21, 5, 19, 36, 13, 28, 18, 7, 31, 18, 19, 34, 15, 14, 27, 27, 11, 41, 31, 11, 68, 16, 10, 71, 30, 20, 23, 21, 16, 40, 40, 13, 57, 37, 23, 37, 24, 16, 67, 44, 16, 41, 20, 20, 54, 55, 12, 43, 54, 15, 81, 26, 15, 65, 34, 37, 50, 20, 29, 70, 68, 14, 52, 46, 14, 79, 43, 18, 60, 70

Offset: 1

Michel Lagneau, Mar 30 2011

nonn

			a(21) = 40 because the polynomial k^2 + k + 41 generates 40 distinct primes for k = 0, 1, .., 39.

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A005846, A056561.

with(numtheory):for n from 1 by 2 to 200 do:m:=0:for k from 0 to n do: x:=k^2+k+n:if
  type(x,prime)=true then m:=m+1:else fi:od:printf(`%d, `,m):od:

A228122 Smallest nonnegative number x such that x^2 + x + 41 has exactly n prime factors counting multiplicities.

Original entry on oeis.org

0, 40, 420, 1721, 14144, 139563, 3019035, 24304266, 206583092, 3838101265

Offset: 1

Shyam Sunder Gupta, Aug 11 2013

			a(1) = 0 because if x = 0 then x^2 + x + 41 = 41, which has 1 prime factor.
a(2) = 40 because if x = 40 then x^2 + x + 41 = 1681 = 41*41, which has 2 prime factors, counting multiplicities.
a(3) = 420 because if x = 420 then x^2 + x + 41 = 176861 = 47*53*71, which has 3 prime factors.

Cf. A005846, A007634, A145292, A145293, A056561.

a = {}; Do[x = 0; While[PrimeOmega[x^2 + x + 41] != k, x++]; AppendTo[a, x], {k, 9}]; a

a(n) = {my(m=0); while (bigomega(m^2+m+41) != n, m++); m;} \\ Michel Marcus, Jan 31 2016

from sympy import factorint
def A228122(n):
    k = 0
    while sum(factorint(k*(k+1)+41).values()) != n:
        k += 1
    return k # Chai Wah Wu, Sep 07 2018

a(9) from Zak Seidov, Feb 01 2016

a(10) from Giovanni Resta, Sep 08 2018

A139219 Primes of the form 41+(n+n^2)/2=41+A000217(n).

Original entry on oeis.org

41, 47, 107, 251, 317, 419, 569, 821, 1031, 1217, 1367, 1637, 1811, 1871, 3527, 5501, 5927, 6257, 6827, 8297, 8819, 9221, 10337, 14747, 17807, 20747, 21569, 22619, 25919, 28961, 31667, 34757, 37991, 43997, 45191, 48869, 49811, 52691, 63587

Offset: 1

Zak Seidov, Apr 11 2008

Corresponding values of n in A139220.

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

Cf. A000217, A056561, A139220, A139221.

[a: n in [0..500] | IsPrime(a) where a is  41 +(n + n^2) div 2]; // Vincenzo Librandi, Mar 22 2013

Select[Table[41+(n+n^2)/2,{n,0,800}],PrimeQ]

A139221 Numbers k such that both 41+(k+k^2)/2 and 41+(k+k^2) are primes.

Original entry on oeis.org

0, 3, 11, 20, 23, 27, 32, 39, 48, 51, 59, 60, 83, 108, 111, 116, 128, 132, 135, 171, 188, 203, 212, 227, 240, 263, 275, 315, 324, 356, 359, 363, 384, 392, 447, 476, 479, 515, 528, 588, 627, 647, 648, 672, 731, 759, 780, 804, 839, 864, 875, 900, 903, 968, 975

Offset: 1

Zak Seidov, Apr 11 2008

nonn

Intersection of A139220 and A056561.

			If k = 11 then 41 + (k + k^2) / 2 = 107 (prime) and 41 + (k + k^2) = 173 (prime).

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A000217, A056561, A139219, A139220.

[k:k in [0..1000]| IsPrime(41+(k+k^2) div 2) and IsPrime(41+k+k^2)]; // Marius A. Burtea, Feb 12 2020

Select[Table[Range[0,2000]],PrimeQ[41+(#+#^2)/2]&&PrimeQ[41+#+#^2]&]
Select[Range[0,1000],AllTrue[41+{(#+#^2)/2,#+#^2},PrimeQ]&] (* Harvey P. Dale, May 21 2024 *)

for(n=0, 1000, if(isprime(binomial(n+1,2) +41) && isprime(n^2+n+41), print1(n", "))) \\ G. C. Greubel, Feb 12 2020

[n for n in (0..1000) if is_prime(binomial(n+1,2)+41) and is_prime(n^2+n+41)] # G. C. Greubel, Feb 12 2020

cp's OEIS Frontend

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

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

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A202018 a(n) = n^2 + n + 41.

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A002837 Numbers k such that k^2 - k + 41 is prime.

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A090102 Leading prime in each set of 7 arising in A090101.

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A259645 Numbers m such that m^2 + 1, 3*m - 1 and m^2 + m + 41 are all prime.

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