A007634 -id:A007634 - cp's OEIS frontend

A055390 Terms of A007634 where n - 40 is not a square.

81, 82, 84, 87, 91, 96, 102, 109, 117, 122, 123, 126, 127, 130, 136, 138, 143, 147, 155, 159, 162, 163, 164, 170, 172, 173, 178, 185, 186, 187, 190, 201, 204, 205, 207, 208, 213, 215, 216, 217, 218, 232, 234, 237, 239, 242, 244, 245, 246, 248, 249, 251, 252

Offset: 1

J. Lowell, Oct 08 2000

nonn

Numbers n such that n^2 + n + 41 is composite and n - 40 is not a square. - Charles R Greathouse IV, Sep 14 2014

Note that if h(n) = n^2 + n + 41, and k(x) = x^2 + 40, then the composition of functions h(k(x)) has an algebraic factorization: h(k(x)) = (x^2 + 40)^2 + (x^2 + 40) + 41 = (x^2 + x + 41)*(x^2 - x + 41). Since both of the expressions in the above product are integers greater than 1, h(k(x)) is composite. - Matt C. Anderson, Oct 24 2012

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

Cf. A007634 (numbers where n^2 + n + 41 is composite). A194634 (numbers in A007634 that are not in 3 parabolas). - Matt C. Anderson, Sep 26 2011

Cf. A201998.

A007634 := {}:
for n from 0 to 1000 do
k := n^2+n+41:
if isprime(k) = false then A007634 := union(A007634, {n}) end if:
end do:
pv1 := Vector(1000, j -> (j-1)^2+40):
p1 := convert(pv1, set):
A055390 := minus(A007634, p1); # Matt C. Anderson, Sep 26 2011
remove(t -> issqr(t-40) or isprime(t^2+t+41), [$1..1000]); # Robert Israel, Nov 24 2017

Select[Range[260],CompositeQ[#^2+#+41]&&!IntegerQ[Sqrt[#-40]]&] (* Harvey P. Dale, Oct 20 2015 *)

is(n)=!isprime(n^2+n+41) && !issquare(n-40) \\ Charles R Greathouse IV, Sep 14 2014

a(n) ~ n. - Charles R Greathouse IV, Sep 14 2014

More terms from David Wasserman, Mar 19 2002

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

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

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

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]

A097823 Numbers n such that n^2+n+41 (Euler's "prime generating polynomial") is not squarefree.

Original entry on oeis.org

40, 603, 798, 890, 917, 1245, 1253, 1318, 1640, 1651, 1721, 2010, 2069, 2251, 2452, 2606, 2649, 3094, 3099, 3321, 3402, 3527, 3607, 4123, 4239, 4301, 4819, 4943, 5002, 5083, 5308, 5372, 5425, 5736, 5790, 5930, 5958, 5998, 6150, 6416, 6511, 6683, 6764

Offset: 1

Hugo Pfoertner, Aug 26 2004

nonn

			a(1)=40: p(40)=40^2+40+41=1681=41^2, a(2)=603: p(603)=364253=197*43^2, a(11)=1721: p(1721)=2963603=43*41^3, a(68)=10428: p(10428)=108753653=743^2*197, a(91)=14144: p(14144)=200066921=47^4*41.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

Cf. A013929 n is not squarefree, A002837 n such that n^2-n+41 is prime, A007634 n such that n^2+n+41 is composite, A005846 primes of form n^2+n+41, A097822 n^2+n+41 has more than 2 prime factors.

Select[Range[10000],!SquareFreeQ[#^2+#+41]&] (* Harvey P. Dale, Nov 06 2011 *)

A145293 a(n) is the smallest nonnegative x such that the Euler polynomial x^2 + x + 41 has exactly n distinct prime proper divisors.

Original entry on oeis.org

0, 41, 420, 2911, 38913, 707864, 6618260, 78776990, 725005500

Offset: 1

Artur Jasinski, Oct 07 2008

The Euler polynomial gives primes for consecutive x from 0 to 39.
For numbers x for which x^2 + x + 41 is not prime, see A007634.
For composite numbers of the form x^2 + x + 41, see A145292.

			a(1)=0 because when x=0 then x^2+x+41=41 (1 distinct prime divisor);
a(2)=41 because when x=41 then x^2+x+41=1763=41*43 (2 distinct prime divisors);
a(3)=420 because when x=420 then x^2+x+41=176861=47*53*71 (3 distinct prime divisors);
a(4)=2911 because when x=2911 then x^2+x+41=8476873=41*47*53*83 (4 distinct prime divisors);
a(5)=38913 because when x=38913 then x^2+x+41=1514260523=43*47*61*71*173 (5 distinct prime divisors);
a(6)=707864 because when x=707864 then x^2+x+41=501072150401=41*43*47*53*71*1607 (6 distinct prime divisors);
a(7)=6618260 because when x=6618260 then x^2+x+41=43801372045901=41*43*47*61*83*131*797 (7 distinct prime divisors);
a(8)=78776990 because when x=78776990 then x^2+x+41=6205814232237131=41*43*61*71*97*131*167*383 (8 distinct prime divisors).
a(9)=725005500: a(9)^2 + a(9) + 41 = 525632975755255541 = 41*43*47*53*61*71*151*397*461. - _Hugo Pfoertner_, Mar 05 2018

Cf. A005846, A007634, A097822, A145292, A145294, A145295.

Cf. A228122. - Zak Seidov, Feb 03 2016

a = {}; Do[x = 1; While[Length[FactorInteger[x^2 + x + 41]] < k - 1, x++ ]; AppendTo[a, x]; Print[x], {k, 2, 10}]; a

Corrected and edited, a(8) added by Zak Seidov, Jan 31 2016
Example for a(8) corrected by Hugo Pfoertner, Mar 02 2018
a(9) from Hugo Pfoertner, Mar 05 2018

A145294 Smallest x >= 0 such that the Euler polynomial x^2 + x + 41 has a prime divisor of multiplicity n.

Original entry on oeis.org

0, 40, 1721, 14144, 2294005, 326924482, 6386359423, 1341160319494, 149759650255065, 1167478867440605, 243422399538851918, 9662500171353620019, 122479951673184550424, 12148820281768361731597, 177497315692809432279207, 11767210525408975519141638

Offset: 1

Artur Jasinski, Oct 07 2008

nonn

The Euler polynomial gives primes for consecutive x from 0 to 39.
For numbers x for which x^2 + x + 41 is not prime, see A007634.
For composite numbers of the form x^2 + x + 41, see A145292.
For the smallest x such that polynomial x^2 + x + 41 has exactly n distinct prime divisors, see A145293.
Sequence interpreted as a(n)^2 + a(n) + 41 having a prime divisor with multiplicity that is exactly n. - Bert Dobbelaere, Jan 22 2019

			a(2)=40 because when x=40 then x^2 + x + 41 = 1681 = 41^2;
a(3)=1721 because when x=1721 then x^2 + x + 41 = 2963603 = 43*41^3;
a(4)=14144 because when x=14144 then x^2 + x + 41 = 200066921 = 41*47^4;
a(5)=2294005 because when x=2294005 then x^2 + x + 41 = 5262461234071 = 35797*43^5.
a(6)=326924482: a(6)^2 + a(6) + 41 = 106879617257892847 = 9915343 * 47^6. - _Hugo Pfoertner_, Mar 08 2018

Bert Dobbelaere, Table of n, a(n) for n = 1..100
Bert Dobbelaere, Python program

Cf. A005846, A007634, A145292, A145293, A145295.

Title changed, a(1) and a(6) from Hugo Pfoertner, Mar 08 2018

More terms from Bert Dobbelaere, Jan 22 2019

A097822 Numbers n such that n^2+n+41 (Euler's "prime generating polynomial") has more than 2 prime factors.

Original entry on oeis.org

420, 431, 491, 492, 514, 533, 573, 574, 603, 614, 655, 686, 738, 775, 798, 858, 861, 890, 895, 901, 904, 917, 919, 942, 984, 989, 1025, 1059, 1116, 1130, 1162, 1169, 1188, 1215, 1222, 1224, 1245, 1251, 1253, 1268, 1271, 1318, 1321, 1334, 1365, 1374, 1407

Offset: 1

Hugo Pfoertner, Aug 26 2004

nonn

All visible sequence terms give exactly 3 prime factors. The smallest composite of the form p(n)=n^2+n+41 with 4 prime factors occurs for p(1721)=2963603=43*41^3. Smallest n with 4 distinct prime factors: p(2911)=8476873=83*53*47*41, smallest n with 5 prime factors: p(14144)=200066921=47^4*41, smallest n with 5 distinct prime factors: p(38913)=1514260523=173*71*61*47*43.

			a(1)=420 because 420^2+420+41=176861=71*53*47 is the first n for which p(n)=n^2+n+41 has more than 2 prime factors. For all smaller n p(n) is either prime or semiprime.

Harvey P. Dale, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

Cf. A002837, A007634, A005846, A097823, A145293.

Select[Range[1500],PrimeOmega[#^2+#+41]>2&] (* Harvey P. Dale, Dec 26 2017 *)

isok(n) = #factor(n^2+n+41)~ > 2; \\ Michel Marcus, Sep 07 2017

Corrected a(19) by Hugo Pfoertner, Sep 07 2017

A201998 Positive numbers n such that n^2 + n + 41 is composite and there are no positive integers c such that n = cx^2 + (c + 1)x + c*41 for an integer x.

Original entry on oeis.org

244, 249, 251, 266, 270, 295, 301, 336, 344, 389, 399, 407, 416, 418, 445, 449, 454, 466, 489, 494, 496, 500, 506, 527, 531, 545, 547, 563, 570, 571, 582, 583, 585, 611, 612, 620, 622, 624, 628, 630, 636, 652, 661, 662, 663, 679, 693, 699

Offset: 1

Matt C. Anderson, Dec 07 2011

nonn

The composition of functions k(x) factors. k(x) = (x^2 + x + 41)*(c^2*x^2 + (c^2 + 2*c)*x + c^2*41 + c + 1). So k(x) is the product of two integers greater than one and thus composite.

John Stillwell, Elements of Number Theory, Springer, 2003, page 3.

Matt C. Anderson A prime producing polynomial writeup

Cf. A007634 (n^2 + n + 41 is composite).

Cf. A235381 (similar to this sequence).

maxn:=1000:
A:={}:
for n from 1 to maxn do
g:=n^2+n+41:
if isprime(g)=false then
A:=A union {n}:
end if:
end do:
# The set A contains values n such that n^2+n+41 is composite and n < maxn.
c:=1:
x:=-1:
p:=41:
q:=c*x^2-(c+1)*x+c*p:
A2:=A:
while q < maxn do
while q < maxn do
A2:=A2 minus {q}:
A2:=A2 minus {c*x^2+(c+1)*x+c*p}:
x:=x+1:
q:=c*x^2-(c+1)*x+c*p:
end do:
c:=c+1:
x:=-1:
q:=c*x^2-(c+1)*x+c*p:
end do:
A2;

Reap[For[n=1, n<700, n++, If[!PrimeQ[n^2+n+41], If[Reduce[c>0 && n == c*x^2+(c+1)*x+41*c , {c, x}, Integers] === False, Sow[n]]]]][[2, 1]] (* Jean-François Alcover, Apr 30 2014 *)

cp's OEIS Frontend

A055390 Terms of A007634 where n - 40 is not a square.

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

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

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

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

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A097823 Numbers n such that n^2+n+41 (Euler's "prime generating polynomial") is not squarefree.

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A201998 Positive numbers n such that n^2 + n + 41 is composite and there are no positive integers c such that n = cx^2 + (c + 1)x + c*41 for an integer x.