A005846 -id:A005846 - cp's OEIS frontend

A272075 Primes of the form k^4 + 29*k^2 + 101.

101, 131, 233, 443, 821, 1451, 2441, 3923, 6053, 9011, 13001, 18251, 25013, 33563, 44201, 57251, 73061, 92003, 114473, 140891, 207371, 295283, 476681, 951491, 1078373, 1369961, 1536251, 1913963, 3472523, 3804341, 4159451, 4943843, 5834531, 7972043, 9925541

Offset: 1

Robert Price, Apr 19 2016

			233 is prime and it is in this sequence since 233 = 2^4 + 29*2^2 + 101.

Robert Price, Table of n, a(n) for n = 1..2264
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A050268, A050267, A005846, A007641, A007635, A048988, A050265, A050266.

Cf. A271980, A272074.

n = Range[0, 100]; Select[#^4 + 29#^2 + 101, PrimeQ[#] &]

lista(nn) = for(n=0, nn, if(ispseudoprime(p=n^4+29*n^2+101), print1(p, ", "))); \\ Altug Alkan, Apr 19 2016

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

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

A164926 Least prime p such that x^2+x+p produces primes for x=0..n-1 and composite for x=n.

Original entry on oeis.org

2, 3, 107, 5, 347, 1607, 1277, 21557, 51867197, 11, 180078317, 1761702947, 8776320587, 27649987598537, 291598227841757, 17

Offset: 1

T. D. Noe, Sep 01 2009

Other known values: a(16)=17 and a(40)=41 (which is generated by Euler's polynomial, A005846). There are no other terms less than 10^12. All of Euler's Lucky numbers, A014556, are in this sequence. Assuming the prime k-tuples conjecture, Mollin's theorem 2.1 shows this sequence is defined for n>0.

a(21)=234505015943235329417 found by J. Waldvogel and Peter Leikauf. [Jens Kruse Andersen, Sep 09 2009]

R. A. Goudsmit, Unusual Prime Number Sequences, Nature, Vol. 214 (June 10, 1967), page 1164.
R. A. Mollin, Prime-Producing Quadratics, The American Mathematical Monthly, Vol. 104, No. 6 (Jun. - Jul., 1997), pp. 529-544.

Cf. A005846, A014556, A067774, A210360, A210361, A210362, A210363, A210364, A210365, A211236, A211237, A211238, A211239, A211240, A354585.

PrimeRun[p_Integer] := Module[{k=0}, While[PrimeQ[k^2+k+p], k++ ]; k]; nn=8; t=Table[0,{nn}]; cnt=0; p=1; While[cnt

a(14) and a(15) from Jens Kruse Andersen, Sep 09 2009

A048059 Primes of the form k^2 + k + 11.

Original entry on oeis.org

11, 13, 17, 23, 31, 41, 53, 67, 83, 101, 167, 193, 251, 283, 317, 353, 431, 563, 661, 823, 881, 941, 1201, 1493, 1571, 1733, 2081, 2267, 2663, 2767, 3203, 3433, 3671, 3793, 3917, 4567, 4703, 5413, 5711, 6173, 6491, 6653, 6983, 7151, 7321, 8753, 8941, 9323, 10111

Offset: 1

N. J. A. Sloane

From Peter Bala, Apr 15 2018: (Start)
The polynomial P(n) := n^2 + n + 11 takes distinct prime values for the 10 consecutive integers n = 0 to 9. It follows that the polynomial P(n-10) = (n - 10)^2 + (n - 10) + 11 takes prime values for the 20 consecutive integers n = 0 to 19, consisting of the 10 primes above each taken twice. We note two consequences of this fact.
1) The polynomial P(2*n-10) = 4*n^2 - 38*n + 101 also takes prime values for the 10 consecutive integers n = 0 to 9.
2)The polynomial P(3*n-10) = 9*n^2 - 57*n + 101 takes prime values for the 7 consecutive integers n = 0 to 6 (= floor(19/3)). In addition, calculation shows that P(3*n-10) also takes prime values for n from -3 to -1. Equivalently put, the polynomial P(3*n-19) = 9*n^2 - 111*n + 353 takes prime values for the 10 consecutive integers n = 0 to 9. Cf. A007635 and A005846. (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.

Cf. A005846, A007635, A048058, A048097, A160548.

[ a: n in [0..200] | IsPrime(a) where a is n^2+n+11 ]; // Vincenzo Librandi, Dec 07 2011

lst={}; Do[p=n^2+n+11; If[PrimeQ[p], AppendTo[lst,p]], {n,0,5*5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 27 2009 *)
Select[Table[n^2+n+11, {n,0,600}], PrimeQ] (* Vincenzo Librandi, Dec 07 2011 *)

a(n) = A048058(A048097(n)). - Elmo R. Oliveira, Apr 20 2025

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]

A272160 Primes of the form abs(8n^2 - 488n + 7243) in order of increasing nonnegative values of n.

Original entry on oeis.org

7243, 6763, 6299, 5851, 5419, 5003, 4603, 4219, 3851, 3499, 3163, 2843, 2539, 2251, 1979, 1723, 1483, 1259, 1051, 859, 683, 523, 379, 251, 139, 43, 37, 101, 149, 181, 197, 197, 181, 149, 101, 37, 43, 139, 251, 379, 523, 683, 859, 1051, 1259, 1483, 1723, 1979

Offset: 1

Robert Price, Apr 21 2016

nonn

			5419 is in this sequence since 8*4^2 - 488*4 + 7243 = 128-1952+7243 = 5419 is prime.

Robert Price, Table of n, a(n) for n = 1..4058
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A050268, A050267, A005846, A007641, A007635, A048988, A050265, A050266.

Cf. A271980, A272074, A272075, A272118, A272159.

n = Range[0, 100]; Select[Abs[8n^2 - 488n + 7243], PrimeQ[#] &]

lista(nn) = for(n=0, nn, if(isprime(p=abs(8*n^2-488*n+7243)), print1(p, ", "))); \\ Altug Alkan, Apr 21 2016

A271144 Primes of the form 42k^3 + 270k^2 - 26436*k + 250703 in order of increasing k.

Original entry on oeis.org

250703, 224579, 199247, 174959, 151967, 130523, 110879, 93287, 77999, 65267, 55343, 48479, 44927, 44939, 48767, 56663, 68879, 85667, 107279, 133967, 165983, 203579, 247007, 296519, 352367, 414803, 484079, 560447, 644159, 735467, 834623, 941879, 1057487

Offset: 1

Robert Price, Apr 23 2016

nonn

			151967 is prime and it is in this sequence since 151967 = 42*4^3 + 270*4^2 - 26436*4 + 250703.

Robert Price, Table of n, a(n) for n = 1..3092
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A050265 - A050268, A005846, A007641, A007635, A048988, A256585

Cf. A271980, A272074, A272075, A272118, A272159, A271143 (associated k).

n = Range[0, 100]; Select[42n^3 + 270n^2 - 26436n + 250703, PrimeQ[#] &]

A272159 Numbers k such that abs(8k^2 - 488k + 7243) 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, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 64, 65, 66, 67, 71

Offset: 1

Robert Price, Apr 21 2016

nonn

From Robert Israel, Apr 21 2016: (Start)
n such that either n <= 61 or 8n^2 - 488n + 7243 is prime.
The first number not in the sequence is 62. (End)

			4 is in this sequence since 8*4^2 - 488*4 + 7243 = 128-1952+7243 = 5419 is prime.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A050268, A050267, A005846, A007641, A007635, A048988, A050265, A050266.

Cf. A271980, A272074, A272075, A272160.

select(n -> isprime(abs(8*n^2 - 488*n + 7243)), [$0..1000]); # Robert Israel, Apr 21 2016

Select[Range[0, 100], PrimeQ[8#^2 - 488# + 7243] &]

lista(nn) = for(n=0, nn, if(isprime(abs(8*n^2-488*n+7243)), print1(n, ", "))); \\ Altug Alkan, Apr 21 2016

A271143 Numbers k such that 42k^3 + 270k^2 - 26436*k + 250703 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, 41, 43, 44, 48, 51, 54, 55, 56, 58, 61, 62, 63, 64, 65, 66, 67, 69, 71, 76, 78, 79, 84, 87, 88, 89, 90, 92

Offset: 1

Robert Price, Apr 23 2016

nonn

40 is the first value not in the sequence.

			4 is in this sequence since 42*4^3 + 270*4^2 - 26436*4 + 250703 = 151967, which is prime.

Robert Price, Table of n, a(n) for n = 1..3092
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A050265-A050268, A005846, A007641, A007635, A048988, A256585, A271980, A272074, A272075, A272160, A271144.

Select[Range[0, 100], PrimeQ[42#^3 + 270#^2 - 26436# + 250703] &]

is(n)=isprime(42*n^3+270*n^2-26436*n+250703) \\ Charles R Greathouse IV, Feb 17 2017

cp's OEIS Frontend

A272075 Primes of the form k^4 + 29*k^2 + 101.

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

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

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A164926 Least prime p such that x^2+x+p produces primes for x=0..n-1 and composite for x=n.

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A048059 Primes of the form k^2 + k + 11.

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

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A272160 Primes of the form abs(8n^2 - 488n + 7243) in order of increasing nonnegative values of n.

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A271144 Primes of the form 42k^3 + 270k^2 - 26436*k + 250703 in order of increasing k.

A272159 Numbers k such that abs(8k^2 - 488k + 7243) is prime.

A271143 Numbers k such that 42k^3 + 270k^2 - 26436*k + 250703 is prime.