A002837 -id:A002837 - cp's OEIS frontend

A007634 Numbers k such that k^2 + k + 41 is composite.

40, 41, 44, 49, 56, 65, 76, 81, 82, 84, 87, 89, 91, 96, 102, 104, 109, 117, 121, 122, 123, 126, 127, 130, 136, 138, 140, 143, 147, 155, 159, 161, 162, 163, 164, 170, 172, 173, 178, 184, 185, 186, 187, 190, 201, 204, 205, 207, 208, 209, 213, 215, 216, 217

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

A subset of this sequence is shown in A055390. - Matt C. Anderson, Jan 05 2014
If prime p divides m^2+m+41, then m+p*j is in the sequence for all j >= 1. - Robert Israel, Nov 24 2017
Euler noted that the first 40 values of this polynomial, starting with k=0, are all primes. - Harvey P. Dale, Jul 25 2023

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Cf. A002837, A055390, A201998.

[n: n in [0..220] | not IsPrime(n^2 + n + 41)]; // Vincenzo Librandi, Sep 28 2012

remove(n -> isprime(n^2+n+41), [$1..1000]); # Robert Israel, Nov 24 2017

Select[Range[220], !PrimeQ[#^2 + # + 41] &] (* Vincenzo Librandi, Sep 28 2012 *)

is(n)=!isprime(n^2+n+41) \\ Charles R Greathouse IV, Apr 25 2014

a(n) ~ n. - Charles R Greathouse IV, Apr 25 2014

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

A331940 Addends k > 0 such that the polynomial x^2 + x + k produces a record of its Hardy-Littlewood Constant.

Original entry on oeis.org

1, 11, 17, 41, 21377, 27941, 41537, 55661, 115721, 239621, 247757

Offset: 1

Hugo Pfoertner, Feb 02 2020

The Hardy and Littlewood Conjecture F provides an estimate of the number of primes generated by a quadratic polynomial P(x) for 0 <= x <= m in the form C * Integral_{x=2..m} 1/log(x) dx, with C given by an Euler product that is a function of the fundamental discriminant of the polynomial. Cohen describes an efficient method for the computation of C.
The following table provides the record values of C, together with the number of primes np generated by the polynomial x^2 + x + a(n) for x <= 10^8 and the actual ratio 2*np/Integral_{x=2..10^8} 1/log(x) dx.
    a(n)    C       np    C from ratio
       1 2.24147  6456835 2.24110
      11 3.25944  9389795 3.25910
      17 4.17466 12027453 4.17460
      41 6.63955 19132653 6.64073
   21377 6.92868 19962992 6.92894
   27941 7.26400 20931145 7.26497
   41537 7.32220 21092134 7.32085
   55661 7.45791 21483365 7.45664
  115721 7.70935 22210771 7.70912
  239621 7.72932 22268336 7.72909
  247757 8.24741 23762118 8.24757
Jacobson and Williams found significantly larger values of C for very large addends k, e.g. C = 2*5.36708 = 10.73416 for k = 3399714628553118047.

Keith Conrad, Hardy-Littlewood Constants. In: Mathematical Properties of Sequences and Other Combinatorial Structures, eds. Jong-Seon No, Hong-Yeop Song, Tor Helleseth, P. Vijay Kumar, Springer New York, 2003, pages 133-154.

Karim Belabas and Henri Cohen, Computation of the Hardy-Littlewood constant for quadratic polynomials, PARI/GP script, 2020.
Henri Cohen, High precision computation of Hardy-Littlewood constants. [Cached pdf version, with permission]
Keith Conrad, Hardy-Littlewood Constants, (2003).
Michael J. Jacobson Jr. and Hugh C. Williams, New Quadratic Polynomials With High Densities Of Prime Values, Math. Comp., 72, 241, 499-519, 2002.

Cf. A002837, A007635, A014556, A116206, A331877.

Cf. A221712, A221713 (Constants C including factor 1/2).

\\ The function HardyLittlewood2 is provided at the Belabas, Cohen link.
hl2max=0; for(add=0,100,my(hl=HardyLittlewood2(n^2+n+add)); if(hl>hl2max,print1(add,", "); hl2max=hl))

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 *)

A060566 a(n) = n^2 - 79*n + 1601.

Original entry on oeis.org

1601, 1523, 1447, 1373, 1301, 1231, 1163, 1097, 1033, 971, 911, 853, 797, 743, 691, 641, 593, 547, 503, 461, 421, 383, 347, 313, 281, 251, 223, 197, 173, 151, 131, 113, 97, 83, 71, 61, 53, 47, 43, 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, 1681

Offset: 0

Jason Earls, Apr 11 2001

a(n) is prime for 0 <= n <= 79. a(80) = 1681 = 41^2.
More than the usual number of terms are shown in order to display the initial 80 primes.
First 80 prime entries are palindromically distributed because a(n) = P(x) = x^2 + x + 41, with x = n - 40 and we observe that P(x) generates primes (A005846) for x = 0 through 39, along with the fact that P(-x) = P(x-1). - Lekraj Beedassy, Apr 24 2006

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 6.
C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Dover Publications, NY, 1966, p. 37, 147.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 115.

Andrew Howroyd, Table of n, a(n) for n = 0..1000 (terms 0..500 from Harry J. Smith and Miquel Cerda)
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002837, A005846, A202018.

[n^2-79*n+1601: n in [0..80]]; // Vincenzo Librandi, Feb 27 2017

A060566:=n->n^2-79*n+1601: seq(A060566(n), n=0..150); # Wesley Ivan Hurt, Jul 10 2017

Table[n^2-79*n+1601,{n,100}] (* or *) LinearRecurrence[{3,-3,1},{1523,1447,1373},100] (* Harvey P. Dale, Jan 14 2017 *)

a(n) = { n^2 - 79*n + 1601 } \\ Harry J. Smith, Jul 07 2009

From Vincenzo Librandi, Feb 27 2017: (Start)
G.f.: (1601 - 3280*x + 1681*x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
a(n) = (n-40)^2 + (n-40) + 41. - Miquel Cerda, Jul 10 2017
E.g.f.: exp(x)*(1601 - 78*x + x^2). - Elmo R. Oliveira, Feb 09 2025

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 16 2007

a(125) in b-file corrected by Andrew Howroyd, Feb 21 2018

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

A273595 Least q > 0 such that min { x >= 0 | q + prime(n)*x + x^2 is composite } is a (local) maximum, cf. A273756 & A273770.

Original entry on oeis.org

43, 47, 53, 71, 83, 113, 131, 173, 251, 281, 383, 461, 503, 593, 743, 73361, 73421, 3071069, 15949847, 76553693, 2204597, 1842719, 246407807, 986578883, 73975907, 4069235123, 1244414939, 25213427, 656856899, 30641069183, 8221946477, 41730358853, 10066886927, 285340609997, 6232338461

Offset: 2

M. F. Hasler, May 26 2016

nonn

This is a subsequence of A273756 which considers all odd numbers (2n+1) instead of only prime(n) as coefficients of the linear term.
All terms are necessarily prime, since this is necessary and sufficient to get a prime for x = 0.
The respective minima (= number of consecutive primes for x = 0, 1, 2, ...) are given in A273597.
It has been pointed out by Don Reble that the prime k-tuple conjecture predicts infinitely long sequences of primes of the given form, therefore we consider the "local" maxima, for q below some appropriate (large) limit: see sequences A273756 & A273770 for further details. - M. F. Hasler, Feb 17 2020

Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.
Index to entries related to primes produced by polynomials.

Cf. A273756, A273770.

Cf. also 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), A097823, A144051, A187057 .. A187060, A190800, A191456 ff.

A273595(n)=A273756(prime(n)\2) \\ changed Feb 17 2020

a(n) = A273756((prime(n) - 1)/2). - M. F. Hasler, Feb 17 2020

Edited and extended using A273756(0..100) due to Don Reble, by M. F. Hasler, Feb 17 2020

A273756 Least p for which min { x >= 0 | p + (2n+1)*x + x^2 is composite } reaches the (local) maximum given in A273770.

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, 73303, 73361, 73421, 73483, 3443897, 3071069, 3071137, 15949847, 76553693, 365462323, 365462399, 2204597, 9721, 1842719, 246407633, 246407719, 246407807, 246407897, 246407989

Offset: 0

M. F. Hasler, May 26 2016

nonn

All terms are prime, since this is necessary and sufficient to get a prime for x = 0.
The values given in A273770 are the number of consecutive primes obtained for x = 0, 1, 2, ....
Sequence A273595 is the subsequence of terms for which 2n+1 is prime.
For even coefficients of the linear term, the answer would always be q=2, the only choice that yields a prime for x=0 and also for x=1 if (coefficient of the linear term)+3 is prime.
The initial term a(n=0) = 41 corresponds to Euler's famous prime-generating polynomial 41+x+x^2. Some subsequent terms are equal to the primes this polynomial takes for x=1,2,3,.... This stems from the fact that adding 2 to the coefficient of the linear term is equivalent to shifting the x-variable by 1. Since here we require x >= 0, we find a reduced subset of the previous sequence of primes, missing the first one, starting with q equal to the second one. (It is known that there is no better prime-generating polynomial of this form than Euler's, see the MathWorld page and A014556. "Better" means a larger p producing p-1 primes in a row. However, the prime k-tuple conjecture suggests that there should be arbitrarily long runs of primes of this form (for much larger p), i.e., longer than 41, but certainly much less than the respective p. Therefore we speak of local maxima.)

Don Reble, Table of n, a(n) for n = 0..100
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial
Index to the OEIS, Entries related to primes produced by polynomials.

Cf. A273595, A273770.
Cf. also 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), A097823, A144051, A187057 ... A187060, A190800, A191456 ff.
The first line of data coincides with that of A202018, A107448, A155884 (and also A140755, A142719, except for some initial terms), which are all related.

A273756(n,p=2*n+1,L=10^(5+n\10),m=0,Q)={forprime(q=1,L, for(x=1,oo, ispseudoprime(q+p*x+x^2)&& next; x>m&& [Q=q,m=x]; break));Q}

Edited, following a remark by Don Reble, by M. F. Hasler, Jan 23 2018

a(27) corrected and more terms from Don Reble, Feb 15 2018

A273770 Max { min { x >= 0 | p + (2n+1)x + x^2 is composite }, p < 10^(5+n/10) }.

Original entry on oeis.org

40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 13, 12, 11, 10, 11, 12, 11, 12, 12, 13, 12, 12, 13, 16, 17, 16, 15, 14, 13, 13, 12, 11, 12, 13, 13, 14, 13, 13, 13, 12, 13, 14, 13, 14, 15, 14, 14, 13, 14, 14, 13

Offset: 0

M. F. Hasler, May 26 2016

nonn

The values for p are given in A273756 which is the main entry, see there for further information and (cross)references.
From the initial values, the sequence seems strictly decreasing, with a(n) = 40-n, however, this property does not persist beyond a(27) = 13.
The upper limit on p ensures that we have a well-defined sequence: The prime k-tuple conjecture predicts existence of arbitrarily long sequences of primes of the given form, and thus unbounded minimal value of x. However, the corresponding prime tuples are expected to appear for much larger values of p. The given limit should be understood as "below the first/next such prime tuple", and in general the values a(n) should not change if that limit would be increased by some orders of magnitude. There might be counterexamples, which would be interesting. The given limit was chosen for lack of a more natural expression, and is relatively small. It could be replaced by a more appropriate function of n if a proposal is available, which should not affect the values given so far. - M. F. Hasler, Jan 22 2018, edited Feb 17 2020

M. F. Hasler, Table of n, a(n) for n = 0..100 (using A273756(0..100) from Don Reble), Feb 17 2020
Index to sequences related to primes produced by polynomials.

Cf. A273595, A273756.

Cf. also A002837, A007634, A005846, A097823, A144051, A187057 ... A187060, A190800, A191456 ff.

{A273770(n, p=2*n+1, L=10^(5+n/10), m)=forprime(q=1,L, for(x=1,oo, ispseudoprime(q+p*x+x^2) || (x>m && !m=x) || break));m}

a(n) = 40 - n for 0 <= n <= 27.

Corrected and extended by Don Reble, Feb 15 2018

A133160 Numbers k such that k^3 + k + 91 is prime.

Original entry on oeis.org

2, 6, 9, 11, 12, 17, 20, 23, 27, 29, 33, 38, 48, 50, 53, 59, 62, 68, 71, 75, 81, 87, 92, 93, 101, 114, 123, 129, 137, 152, 153, 170, 171, 176, 186, 197, 198, 222, 225, 246, 248, 254, 258, 263, 270, 284, 288, 293, 296, 297, 303, 306, 309, 314, 321, 324, 332, 335

Offset: 1

Parthasarathy Nambi, Dec 17 2007

			2 is in the sequence because 2^3+2+91 = 101 is prime.
93 is in the sequence because 93^3+93+91 = 804541 is prime.

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

Cf. A002837.

[ n: n in [0..400] | IsPrime(n^3+n+91) ]; // Vincenzo Librandi, Jan 31 2011

a:=proc(n) if isprime(n^3+n+91)=true then n else end if end proc: seq(a(n),n =1..300); # Emeric Deutsch, Jan 08 2008

Select[Range[400], PrimeQ[ #^3 + # + 91] &] (* Stefan Steinerberger, Dec 24 2007 *)

is(n)=isprime(n^3+n+91) \\ Charles R Greathouse IV, Jun 06 2017

More terms from Stefan Steinerberger, Dec 24 2007

cp's OEIS Frontend

A007634 Numbers k such that k^2 + k + 41 is composite.

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

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A331940 Addends k > 0 such that the polynomial x^2 + x + k produces a record of its Hardy-Littlewood Constant.

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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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A060566 a(n) = n^2 - 79*n + 1601.

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

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A273595 Least q > 0 such that min { x >= 0 | q + prime(n)*x + x^2 is composite } is a (local) maximum, cf. A273756 & A273770.

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A273770 Max { min { x >= 0 | p + (2n+1)x + x^2 is composite }, p < 10^(5+n/10) }.