A005846 -id:A005846 - cp's OEIS frontend

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

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

A116206 Primes of the form n^2 + n + 55661, with n >= 0.

Original entry on oeis.org

55661, 55663, 55667, 55673, 55681, 55691, 55717, 55733, 55793, 55817, 55843, 55871, 55901, 55933, 55967, 56003, 56041, 56081, 56123, 56167, 56311, 56417, 56473, 56531, 56591, 56783, 56921, 56993, 57143, 57221, 57301, 57383, 57467

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Apr 14 2007

n^2+n+55661 is the first or second most efficient formula for finding primes until we get to n^2+n+154001 (competitors are n^2+n+41, n^2+n+55661, n^2+n+41537, n^2+n+27941, n^2+n+21377, ...)

Cf. A005846.

A117624 Primes of the form f(k) = 9k^6 - 804k^5 + 29836k^4 - 588615k^3 + 6509950k^2 - 38263500k + 93363947 for values of k >= 0.

Original entry on oeis.org

93363947, 61050823, 38620051, 23498297, 13649371, 7493947, 3835763, 1794301, 743947, 259631, 68947, 10753, 251, 547, 691, 197, 43, 151, 3347, 25801, 113947, 367883, 971251, 2227597, 4603211, 8776447, 15693523, 26630801, 102768947, 218611051, 1738931741

Offset: 1

Parviz Afereidoon (afereidoon(AT)gmail.com), Apr 08 2006

This polynomial f(n) generates 28 consecutive prime numbers for n = 0 to n = 27.

In n^2 + n + 41, substitute n -> 3*n^3 - 134*n^2 + 1980*n - 9663.

			f(1) = 9(1)^6 - 804(1)^5 + 29836(1)^4 - 588615(1)^3 + 6509950(1)^2 - 38263500(1) + 93363947 = 61050823, a prime number.

P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 137.

Vincenzo Librandi, Table of n, a(n) for n = 1..2825
Carlos Rivera, Puzzle 232. Primes and Cubic polynomials, The Prime Puzzles & Problems Connection.
Eric Weisstein's World of Mathematics, Prime-generating polynomial.

Cf. A005846.

[a: n in [0..200] | IsPrime(a) where a is 9*n^6 - 804*n^5 + 29836*n^4 - 588615*n^3 + 6509950*n^2 - 38263500*n + 93363947 ]; // Vincenzo Librandi, Jul 28 2025

f[n_] := 9n^6-804n^5+29836n^4-588615n^3+6509950n^2-38263500n+93363947; f[Select[Range[0, 100], PrimeQ[f[ # ]] &]] (* Stefan Steinerberger, Apr 16 2006 *)

Edited by Don Reble, Apr 14 2006

More terms from Petros Hadjicostas, Nov 04 2019

A145202 Primes of form 4n^2 + 4n + 653.

Original entry on oeis.org

653, 661, 677, 701, 733, 773, 821, 877, 941, 1013, 1093, 1181, 1277, 1381, 1493, 1613, 1741, 1877, 2333, 2677, 2861, 3253, 3461, 3677, 4133, 4373, 4621, 4877, 5413, 5693, 5981, 6277, 6581, 7213, 7541, 7877, 8221, 8573, 8933, 9677, 10061, 10453, 10853

Offset: 1

Klaus Brockhaus, Oct 04 2008

nonn

First 18 terms are for n from 0 through 17, next terms are for n = 20, 22, 23, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 37, 38, 40, ...
The sequence of n such that 4*n^2 + 4*n + 653 is composite starts 18, 19, 21, 24, 28, 33, 39, 46, 54, 60, 61, 62, 63, 65, 67, 72, 73, 75, 81, 82, 84, 85, 86, 93, 95, 96, 100, ...
These primes are in A000414. [Bruno Berselli, Apr 20 2014]

			a(18) = 4*17^2 + 4*17 + 653 = 1877.

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

A145125 is essentially the same sequence.

Cf. A005846 (primes of form n^2 + n + 41).

[a: n in [0..100] | IsPrime(a) where a is  4*n^2 + 4*n + 653]; // Vincenzo Librandi, Apr 21 2014

Select[Table[4 n^2 + 4 n + 653, {n, 0, 100}], PrimeQ] (* Vincenzo Librandi, Apr 21 2014 *)

{for(n=0, 50, if(isprime(p=4*n^2+4*n+653), print1(p, ",")))}

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

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

A267252 Primes of the form abs(103n^2 - 4707n + 50383) in order of increasing nonnegative n.

Original entry on oeis.org

50383, 45779, 41381, 37189, 33203, 29423, 25849, 22481, 19319, 16363, 13613, 11069, 8731, 6599, 4673, 2953, 1439, 131, 971, 1867, 2557, 3041, 3319, 3391, 3257, 2917, 2371, 1619, 661, 503, 1873, 3449, 5231, 7219, 9413, 11813, 14419, 17231, 20249, 23473, 26903

Offset: 1

Robert Price, Apr 28 2016

This polynomial is a transformed version of the polynomial P(x) = 103*x^2 + 31*x - 3391 whose absolute value gives 43 distinct primes for -23 <= x <= 19, found by G. W. Fung in 1988. - Hugo Pfoertner, Dec 13 2019

			33203 is in this sequence since 103*4^2 - 4707*4 + 50383  = 1648-18828+50383 = 33203 is prime.

Paulo Ribenboim, The Little Book of Bigger Primes, Second Edition, Springer-Verlag New York, 2004.

Robert Price, Table of n, a(n) for n = 1..3885
François Dress and Michel Olivier, Polynômes prenant des valeurs premières, Experimental Mathematics, Volume 8, Issue 4 (1999), 319-338.
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

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

Cf. A271980, A272030, A272074, A272075, A272118, A272159, A271143, A272284, A272323, A267069, A330363.

n = Range[0, 100]; Abs @ Select[103n^2 - 4707n + 50383 , PrimeQ[#] &]

lista(nn) = for(n=0, nn, if(isprime(p=abs(103*n^2-4707*n+50383)), print1(p, ", "))); \\ Altug Alkan, Apr 28 2016, corrected by Hugo Pfoertner, Dec 13 2019

Title corrected by Hugo Pfoertner, Dec 13 2019

A268200 Nonnegative numbers n such that abs(n^4 - 97n^3 + 3294n^2 - 45458n + 213589) 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, 51, 52, 53, 54, 62, 65, 67, 70, 72, 73, 74, 75, 84, 85, 86, 90, 92

Offset: 1

Robert Price, Apr 30 2016

nonn

50 is the smallest number not in this sequence.

			4 is in this sequence since abs(4^4 - 97*4^3 + 3294*4^2 - 45458*4 + 213589) = abs(256-6208+52704-181832+213589) = 78509 is prime.

Robert Price, Table of n, a(n) for n = 1..2676
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A050268, A050267, A005846, A007641, A007635, A048988, A050265, A050266, A271980, A272030, A272074, A272075, A272160, A271144, A272285, A272401, A272438, A272444, A272410.

Select[Range[0, 100], PrimeQ[#^4 - 97#^3 + 3294#^2 - 45458# + 213589] &]

is(n)=isprime(abs(n^4-97*n^3+3294*n^2-45458*n+213589)) \\ Charles R Greathouse IV, Feb 20 2017

cp's OEIS Frontend

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

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A116206 Primes of the form n^2 + n + 55661, with n >= 0.

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A117624 Primes of the form f(k) = 9*k^6 - 804*k^5 + 29836*k^4 - 588615*k^3 + 6509950*k^2 - 38263500*k + 93363947 for values of k >= 0.

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A145202 Primes of form 4*n^2 + 4*n + 653.

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

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Maple

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

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Python

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

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Magma

Maple

Mathematica

PARI

A257362 Odd primes modulo which -163 is a square.

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A117624 Primes of the form f(k) = 9k^6 - 804k^5 + 29836k^4 - 588615k^3 + 6509950k^2 - 38263500k + 93363947 for values of k >= 0.

A145202 Primes of form 4n^2 + 4n + 653.

A267252 Primes of the form abs(103n^2 - 4707n + 50383) in order of increasing nonnegative n.