A060566 -id:A060566 - cp's OEIS frontend

A073104 Duplicate of A060566.

1523, 1447, 1373, 1301, 1231, 1163, 1097, 1033, 971, 911, 853, 797, 743, 691, 641

Offset: 1

dead

A202018 a(n) = n^2 + n + 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, 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

A211773 Prime-generating polynomial: a(n) = 2n^2 - 108n + 1259.

Original entry on oeis.org

1259, 1153, 1051, 953, 859, 769, 683, 601, 523, 449, 379, 313, 251, 193, 139, 89, 43, 1, -37, -71, -101, -127, -149, -167, -181, -191, -197, -199, -197, -191, -181, -167, -149, -127, -101, -71, -37, 1, 43, 89, 139, 193, 251, 313, 379, 449, 523, 601, 683, 769, 859, 953

Offset: 0

Marius Coman, May 18 2012

This polynomial generates 92 primes (66 distinct ones) for 0 <= n <= 99 (in fact the next two terms are still primes but we keep the range 0-99, customary for comparisons), just three primes less than the record held by Euler's polynomial for n = m - 35, which is m^2 - 69*m + 1231 (see the link below), but having six distinct primes more than this one.
The nonprime terms in the first 100 are: 1 (taken twice), 1369 = 37^2, 1849 = 43^2, 4033 = 37*109, 5633 = 43*131, 7739 = 71*109 and 8251 = 37*223.
For n = 2*m - 34 we obtain the polynomial 8*m^2 - 488*m + 7243, which generates 31 primes in a row starting from m = 0 (polynomial already reported, see the link below).
For n = 4*m - 34 we obtain the polynomial 32*m^2 - 976*m + 7243, which generates 31 primes in row starting from m = 0.
The polynomial 2*n^2 + 40*n + 1, which generates the positive terms of this sequence in ascending order (i.e., a(37), ...), yields 10774009 distinct primes for 0 <= n < 49999999 while Euler's polynomial (n^2 - n + 41) gives 9967520 primes in same range. - Mikk Heidemaa, Feb 23 2016

Joe L. Mott and Kermite Rose, Prime-Producing Cubic Polynomials in Lecture Notes in Pure and Applied Mathematics (Vol. 220), Marcel Dekker Inc., 2001, pages 281-317.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Marius Coman, Ten prime-generating quadratic polynomials, Preprint 2015.
Joe L. Mott and Kermite Rose, Prime-Producing Cubic Polynomials.
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A060566, A202018.

[2*n^2-108*n+1259: n in [0..49]]; // Bruno Berselli, May 18 2012

Table[2 n^2 + 40 n + 1, {n, -37, 962}] (* Mikk Heidemaa, Feb 18 2016 *)

a(n)=2*n^2 - 108*n + 1259 \\ Charles R Greathouse IV, Jun 29 2017

G.f.: (1259 - 2624*x + 1369*x^2)/(1-x)^3. - Bruno Berselli, May 18 2012
a(n-37) = 2*n^2 + 40*n + 1. - Mikk Heidemaa, Feb 18 2016
From Elmo R. Oliveira, Feb 09 2025: (Start)
E.g.f.: exp(x)*(1259 - 106*x + 2*x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A212325 Prime-generating polynomial: a(n) = n^2 + 3*n - 167.

Original entry on oeis.org

-167, -163, -157, -149, -139, -127, -113, -97, -79, -59, -37, -13, 13, 41, 71, 103, 137, 173, 211, 251, 293, 337, 383, 431, 481, 533, 587, 643, 701, 761, 823, 887, 953, 1021, 1091, 1163, 1237, 1313, 1391, 1471, 1553, 1637, 1723, 1811, 1901, 1993, 2087, 2183, 2281, 2381

Offset: 0

Marius Coman, May 14 2012

The polynomial generates 24 primes in absolute value (23 distinct ones) in row starting from n=0 (and 42 primes in absolute value for n from 0 to 46).
The polynomial n^2 - 49*n + 431 generates the same primes in reverse order.
Note: we found in the same family of prime-generating polynomials (with the discriminant equal to 677) the polynomial 13*n^2 - 311*n + 1847 (13*n^2 - 469*n + 4217) generating 23 primes and two noncomposite numbers (in absolute value) in row starting from n=0 (1847, 1549, 1277, 1031, 811, 617, 449, 307, 191, 101, 37, -1, -13, 1, 41, 107, 199, 317, 461, 631, 827, 1049, 1297, 1571, 1871).
Note: another interesting algorithm to produce prime-generating polynomials could be N = m*n^2 + (6*m+1)*n + 8*m + 3, where m, 6*m+1 and 8*m+3 are primes. For m=7 then n=t-20 we get N = 7*t^2 - 237*t + 1999, which generates the following primes: 239, 163, 101, 53, 19, -1, -7, 1, 23, 59, 109, 173, 251 (we can see the same pattern: …, -1, -m, 1, …).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A060566 (an 80 primes generating pol.), A202018 (Euler's p.g.p.), A050268, A181963, A181973, A182409, A211773, A318791, A320772, A330363 (other p.g.p.).

[n^2+3*n-167: n in [0..47]]; // Bruno Berselli, May 18 2012

Table[n^2+3n-167,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{-167,-163,-157},50] (* Harvey P. Dale, Feb 08 2020 *)

Vec((-167+338*x-169*x^2)/(1-x)^3+O(x^99)) \\ Charles R Greathouse IV, Oct 01 2012

apply( {A212325(n)=(n+3)*n-167}, [0..55]) \\ M. F. Hasler, Feb 11 2025

def A212325(n=None, upto=None): return(A212325(i)for i in range(n or 0, upto or 2**63)) if upto or n is None else(n+3)*n-167 # M. F. Hasler, Feb 11 2025

G.f.: (-167 + 338*x - 169x^2)/(1-x)^3. - Bruno Berselli, May 18 2012
From Elmo R. Oliveira, Feb 10 2025: (Start)
E.g.f.: exp(x)*(-167 + 4*x + x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

Edited by Bruno Berselli, May 18 2012

A155884 a(n) = n^2 - n + 41 if this is a prime, a(n) = a(n-40) otherwise.

Original entry on oeis.org

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, 41, 43, 1847, 1933, 61, 2111, 2203, 2297, 2393, 131

Offset: 0

Roger L. Bagula and M. F. Hasler, Jan 29 2009

It is well known that for 0 <= n <= 40, the polynomial f(n) = n^2 - n + 41 does yield a prime number, so the sequence is well defined.
A variant of A005846, A060566, A142719. All these aim at extending the series of prime values of Euler's famous prime-producing polynomial P(n) = n^2 + n + 41, see references in A005846. [The present sequence considers f(n) = P(n-1) which is completely equivalent.]
The present sequence is a simplification of an extended variant of A142719. By construction, all terms of the present sequence are prime, but in contrast to A005846, prime values of the polynomial remain at the "correct" position, a(n) = f(n). The "substituted" values are easily recognized as they follow local maxima. Of course one could equally well insert a(n) = 2 whenever f(n) is composite.
The present sequence contains only primes. A different sequence, defined by "a(n) = f(n) if this is prime, a(n) = f(n-40) otherwise", does not always produce primes.

Cf. A005846, A060566, A142719.

a(n) = { while( !isprime( n^2-n+41 ), n-=40 ); n^2-n+41 }

A226097 a(n) = ((-1)^n + 2n - 38)(2*n - 38) + 41.

Original entry on oeis.org

1447, 1373, 1163, 1097, 911, 853, 691, 641, 503, 461, 347, 313, 223, 197, 131, 113, 71, 61, 43, 41, 47, 53, 83, 97, 151, 173, 251, 281, 383, 421, 547, 593, 743, 797, 971, 1033, 1231, 1301, 1523, 1601, 1847, 1933, 2203, 2297, 2591, 2693, 3011, 3121, 3463, 3581, 3947

Offset: 0

Arkadiusz Wesolowski, May 26 2013

a(n) are distinct primes for n = 0 to 59.

All terms are in A202018.

Arkadiusz Wesolowski, Table of n, a(n) for n = 0..1000
Carlos Rivera, Puzzle 782
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000040, A005846, A060566, A202018.

[((-1)^n+a)*a+41 where a is 2*n-38 : n in [0..50]];

g[n_] := 2*n - 38; f[n_] := ((-1)^n + g[n])*g[n] + 41; Table[f[n], {n, 0, 50}]
EulerP[n_] := n^2 - n + 41; f[n_] := 2*n - (3 + (-1)^n)/2; LinearRecurrence[{1, 2, -2, -1, 1}, Table[EulerP@f[n], {n, 19, 15, -1}], {0, 50}]

Vec((1447 - 74*x - 3104*x^2 + 82*x^3 + 1681*x^4) / ((1 - x)^3*(1 + x)^2) + O(x^100)) \\ Colin Barker, Aug 14 2017

G.f.: (1447-2*x*(37+1552*x-41*x^2)+(41*x^2)^2)/((1+x)^2*(1-x)^3).
From Colin Barker, Aug 14 2017: (Start)
G.f.: (1447 - 74*x - 3104*x^2 + 82*x^3 + 1681*x^4) / ((1 - x)^3*(1 + x)^2).
a(n) = 4*n^2 - 150*n + 1447 for n even.
a(n) = 4*n^2 - 154*n + 1523 for n odd.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
(End)

A289762 Triangular array T(m,k) = (m+1-k)^2 + k - 1 with m (row) >= 1 and k (column) >= 1, read by rows.

Original entry on oeis.org

1, 1, 4, 2, 2, 4, 9, 5, 3, 3, 5, 9, 16, 10, 6, 4, 4, 6, 10, 16, 25, 17, 11, 7, 5, 5, 7, 11, 17, 25, 36, 26, 18, 12, 8, 6, 6, 8, 12, 18, 26, 36, 49, 37, 27, 19, 13, 9, 7, 7, 9, 13, 19, 27, 37, 49, 64, 50, 38, 28, 20, 14, 10, 8, 8, 10, 14, 20, 28, 38, 50, 64, 81, 65, 51, 39, 29, 21, 15, 11, 9

Offset: 1

Miquel Cerda, Jul 12 2017

The n-th row is of length = max(2n, 1) and the row sum is (2n^3 + 6n^2 - 2n) / 3.

Rows m = 2, 3, 5, 11, and 41 (Euler's lucky numbers) give the prime numbers generated by the famous polynomials, but twice each one between m^2.

			The m-th row start and end: T(m,1) = m^2, ..., T(m,2m) = m^2.
In general T(m,k) = T(m,2m+1-k).
m\k    1     2     3     4     5     6     7     8     9     10
1      1,    1,
2      4,    2,    2,    4
3      9,    5,    3,    3,    5,    9
4      16,   10,   6,    4,    4,    6,    10,   16
5      25,   17,   11,   7,    5,    5,    7,    11,   17,   25
6      36,   26,   18,   12,   8,    6,    6,    8,    12,   18, ...
7      49,   37,   27,   19,   13,   9,    7,    7,    9,    13, ...
8      64,   50,   38,   28,   20,   14,   10,   8,    8,    10, ...
9      81,   65,   51,   39,   29,   21,   15,   11,   9,    9, ...
10     100,  82,   66,   52,   40,   30    22,   16,   12,   10, ...
The T(m,k) sequence as an isosceles triangle:
                                     1  1
                                 4   2  2  4
                             9   5   3  3  5  9
                         16  10  6   4  4  6  10  16
                     25  17  11  7   5  5  7  11  17  25
                 36  26  18  12  8   6  6  8  12  18  26  36
             49  37  27  19  13  9   7  7  9  13  19  27  37  49
         64  50  38  28  20  14  10  8  8  1  14  20  28  38  50  64
     81  65  51  39  29  21  15  11  9  9  11 15  21  29  39  51  65  81
100  82  66  52  40  30  22  16  12  10 10 12 16  22  30  40  52  66  82  100

Miquel Cerda, Table of n, a(n) for n = 1..306
Miquel Cerda, Triangle rows 1..41
Miquel Cerda, Isosceles triangle Rows 1..41

m(41, k+1) = A060566(n), left and right border gives A000290(n).

Table[(m + 1 - k)^2 + k - 1, {m, 0, 10}, {k, 2 m}] /. {} -> {0} // Flatten (* Michael De Vlieger, Jul 12 2017 *)

T(m,k) = (m+1-k)^2+k-1 \\ Charles R Greathouse IV, Jul 12 2017

The formula that gives the integers in the m-th rows can be expressed using quadratic polynomials:
for row m = 1, a(k) = k^2 - 3*k + 3
for row m = 2, a(k) = k^2 - 5*k + 8
for row m = 3, a(k) = k^2 - 7*k + 15
for row m = 4, a(k) = k^2 - 9*k + 24
for row m = 5, a(k) = k^2 - 11*k + 35
for row m = 6, a(k) = k^2 - 13*k + 48
etc.

cp's OEIS Frontend

A073104 Duplicate of A060566.

Views

Author

Keywords

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Haskell

Magma

Maple

Mathematica

PARI

Sage

Scala

Formula

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A211773 Prime-generating polynomial: a(n) = 2*n^2 - 108*n + 1259.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A212325 Prime-generating polynomial: a(n) = n^2 + 3*n - 167.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Python

Formula

Extensions

A155884 a(n) = n^2 - n + 41 if this is a prime, a(n) = a(n-40) otherwise.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A226097 a(n) = ((-1)^n + 2*n - 38)*(2*n - 38) + 41.

Views

Author

Keywords

Comments

Links

Crossrefs

A211773 Prime-generating polynomial: a(n) = 2n^2 - 108n + 1259.

A226097 a(n) = ((-1)^n + 2n - 38)(2*n - 38) + 41.