A033217 -id:A033217 - cp's OEIS frontend

A106856 Primes of the form x^2 + xy + 2y^2, with x and y nonnegative.

2, 11, 23, 37, 43, 53, 71, 79, 107, 109, 127, 137, 149, 151, 163, 193, 197, 211, 233, 239, 263, 281, 317, 331, 337, 373, 389, 401, 421, 431, 443, 463, 487, 491, 499, 541, 547, 557, 569, 599, 613, 617, 641, 653, 659, 673, 683, 739, 743, 751, 757, 809, 821

Offset: 1

T. D. Noe, May 09 2005, Apr 28 2008

Discriminant=-7. Binary quadratic forms ax^2 + bxy + cy^2 have discriminant d = b^2 - 4ac.

Consider sequences of primes produced by forms with -100

The Mathematica function QuadPrimes2 is useful for finding the primes less than "lim" represented by the positive definite quadratic form ax^2 + bxy + cy^2 for any a, b and c satisfying a>0, c>0, and discriminant d<0. It does this by examining all x>=0 and y>=0 in the ellipse ax^2 + bxy + cy^2 <= lim. To find the primes generated by positive and negative x and y, compute the union of QuadPrimes2[a,b,c,lim] and QuadPrimes2[a,-b,c,lim]. - T. D. Noe, Sep 01 2009

For other programs see the "Binary Quadratic Forms and OEIS" link.

David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.
L. E. Dickson, History of the Theory of Numbers, Vol. 3, Chelsea, 1923.

Zak Seidov and N. J. A. Sloane, Table of n, a(n) for n = 1..10000 (The first 1225 terms were found by Zak Seidov)
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Discriminants in the range -3 to -100: A007645 (d=-3), A002313 (d=-4), A045373, A106856 (d=-7), A033203 (d=-8), A056874, A106857 (d=-11), A002476 (d=-12), A033212, A106858-A106861 (d=-15), A002144, A002313 (d=-16), A106862-A106863 (d=-19), A033205, A106864-A106865 (d=-20), A106866-A106869 (d=-23), A033199, A084865 (d=-24), A002476, A106870 (d=-27), A033207 (d=-28), A033221, A106871-A106874 (d=-31), A007519, A007520, A106875-A106876 (d=-32), A106877-A106881 (d=-35), A040117, A068228, A106882 (d=-36), A033227, A106883-A106888 (d=-39), A033201, A106889 (d=-40), A106890-A106891 (d=-43), A033209, A106282, A106892-A106893 (d=-44), A033232, A106894-A106900 (d=-47), A068229 (d=-48), A106901-A106904 (d=-51), A033210, A106905-A106906 (d=-52), A033235, A106907-A106913 (d=-55), A033211, A106914-A106917 (d=-56), A106918-A106922 (d=-59), A033212, A106859 (d=-60), A106923-A106930 (d=-63), A007521, A106931 (d=-64), A106932-A106933 (d=-67), A033213, A106934-A106938 (d=-68), A033246, A106939-A106948 (d=-71), A106949-A106950 (d=-72), A033212, A106951-A106952 (d=-75), A033214, A106953-A106955 (d=-76), A033251, A106956-A106962 (d=-79), A047650, A106963-A106965 (d=-80), A106966-A106970 (d=-83), A033215, A102271, A102273, A106971-A106974 (d=-84), A033256, A106975-A106983 (d=-87), A033216, A106984 (d=-88), A106985-A106989 (d=-91), A033217 (d=-92), A033206, A106990-A107001 (d=-95), A107002-A107008 (d=-96), A107009-A107013 (d=-99).
Other collections of quadratic forms: A139643, A139827.
For a more comprehensive list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
Cf. also A242660.

QuadPrimes2[a_, b_, c_, lmt_] := Module[{p, d, lst = {}, xMax, yMax}, d = b^2 - 4a*c; If[a > 0 && c > 0 && d < 0, xMax = Sqrt[lmt/a]*(1+Abs[b]/Floor[Sqrt[-d]])]; Do[ If[ 4c*lmt + d*x^2 >= 0, yMax = ((-b)*x + Sqrt[4c*lmt + d*x^2])/(2c), yMax = 0 ]; Do[p = a*x^2 + b*x*y + c*y^2; If[ PrimeQ[ p]  && p <= lmt && !MemberQ[ lst, p], AppendTo[ lst, p]], {y, 0, yMax}], {x, 0, xMax}]; Sort[ lst]];
QuadPrimes2[1, 1, 2, 1000]
(This is a corrected version of the old, incorrect, program QuadPrimes. - N. J. A. Sloane, Jun 15 2014)
max = 1000; Table[yy = {y, 1, Floor[Sqrt[8 max - 7 x^2]/4 - x/4]}; Table[ x^2 + x y + 2 y^2, yy // Evaluate], {x, 0, Floor[Sqrt[max]]}] // Flatten // Union // Select[#, PrimeQ]& (* Jean-François Alcover, Oct 04 2018 *)

list(lim)=my(q=Qfb(1,1,2), v=List([2])); forprime(p=2, lim, if(vecmin(qfbsolve(q, p))>0, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Aug 05 2016

Removed old Mathematica programs - T. D. Noe, Sep 09 2009

Edited (pointed out error in QuadPrimes, added new version of program, checked and extended b-file). - N. J. A. Sloane, Jun 06 2014

A028929 Numbers represented by quadratic form with Gram matrix [ 4, 1; 1, 6 ], divided by 2.

Original entry on oeis.org

0, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18, 24, 26, 27, 29, 31, 32, 36, 39, 41, 46, 47, 48, 50, 52, 54, 58, 62, 64, 69, 71, 72, 73, 75, 78, 81, 82, 87, 92, 93, 94, 96, 98, 100, 104, 108, 116, 117, 118, 123, 124, 127, 128, 131, 138, 139, 141, 142, 144, 146, 147, 150, 151, 156

Offset: 1

N. J. A. Sloane

Nonnegative integers of the form 2x^2 + x*y + 3y^2, a positive definite quadratic form of discriminant -23. - N. J. A. Sloane, Jul 09 2014

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

For primes see A106867. Cf. A028958, A033217.

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Mar 29 2000

A104217 Period of Perrin (0,2,3,2,5,5,..., A001608) sequence mod n.

Original entry on oeis.org

1, 7, 13, 14, 24, 91, 48, 28, 39, 168, 120, 182, 183, 336, 312, 56, 288, 273, 180, 168, 624, 840, 22, 364, 120, 1281, 117, 336, 871, 2184, 993, 112, 1560, 2016, 48, 546, 1368, 1260, 2379, 168, 1723, 4368, 231, 840, 312, 154, 2257, 728, 336, 840, 3744, 2562

Offset: 1

Anthony C Robin, Mar 14 2005

nonn

Analogy to A001175, Pisano periods (or Pisano numbers): period of Fibonacci numbers mod n AND to A046738 for Perrin sequence, where a(n)=a(n-2)+a(n-3)

It appears that the n such that n-1 divides a(n) is the set of primes of the form x^2+23*y^2 (A033217). The discriminant of the characteristic polynomial of the Perrin sequence is -23. - T. D. Noe, Feb 23 2007

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A001175, A046738 and Perrin sequence A001608.

Table[a={0,2,3}; a=a0=Mod[a, n]; k=0; While[k++; s=a[[2]]+a[[1]]; a=RotateLeft[a]; a[[ -1]]=Mod[s,n]; a!=a0]; k, {n,100}] (* T. D. Noe, Oct 10 2006 *)

from math import lcm
from functools import lru_cache
from sympy import factorint
@lru_cache(maxsize=None)
def A104217(n):
    if n < 4:
        return (1,7,13)[n-1]
    f = factorint(n).items()
    if len(f) > 1:
        return lcm(*(A104217(a**b) for a,b in f))
    else:
        k,x = 1, (0,2,3)
        while x != (3,0,2):
            k += 1
            x = (x[1], x[2], (x[0]+x[1]) % n)
        return k # Chai Wah Wu, Apr 25 2025

More terms from T. D. Noe, Oct 10 2006

A028958 Numbers represented by quadratic form with Gram matrix [ 2, 1; 1, 12 ] (divided by 2).

Original entry on oeis.org

0, 1, 4, 6, 8, 9, 12, 16, 18, 23, 24, 25, 26, 27, 32, 36, 39, 48, 49, 52, 54, 58, 59, 62, 64, 72, 78, 81, 82, 87, 92, 93, 94, 96, 100, 101, 104, 108, 116, 117, 121, 123, 124, 128, 138, 141, 142, 144, 146, 150, 156, 162, 164, 167, 169, 173, 174, 184, 186, 188, 192

Offset: 1

N. J. A. Sloane

nonn

Nonnegative integers of the form x^2 + x*y + 6*y^2, discriminant -23. - Ray Chandler, Jul 12 2014
The Gram matrix is positive-definite, therefore, if w := (1 + sqrt(-23)) / 2, then |x + w*y|^2 = x^2 + x*y + 6*y^2 > 0 for all integers x and y except x = y = 0. - Michael Somos, Mar 28 2015
The theta function of the lattice with basis [1, w] is the g.f. of A028959, therefore, A028959(n) is positive if and only if n is in this sequence. - Michael Somos, Mar 28 2015

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

For primes see A033217. Cf. A028929, A106867.

Cf. A028959.

More terms from Larry Reeves (larryr(AT)acm.org), Mar 29 2000

A309138 Nonnegative integers of the form x^2 + 23*y^2.

Original entry on oeis.org

0, 1, 4, 9, 16, 23, 24, 25, 27, 32, 36, 39, 48, 49, 59, 64, 72, 81, 87, 92, 93, 96, 100, 101, 104, 108, 117, 121, 123, 128, 141, 144, 156, 167, 169, 173, 192, 196, 207, 208, 211, 213, 216, 219, 223, 225, 232, 236, 243, 248, 256, 261, 271, 279, 288, 289, 307, 312

Offset: 1

Hugo Pfoertner, Jul 14 2019

nonn

Discriminant of positive definite binary quadratic form: -92.

Peter Luschny, Binary Quadratic Forms
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Primes in this sequence: A033217.

Cf. A028958.

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cp's OEIS Frontend

A106856 Primes of the form x^2 + xy + 2y^2, with x and y nonnegative.

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A028929 Numbers represented by quadratic form with Gram matrix [ 4, 1; 1, 6 ], divided by 2.

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A104217 Period of Perrin (0,2,3,2,5,5,..., A001608) sequence mod n.

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A028958 Numbers represented by quadratic form with Gram matrix [ 2, 1; 1, 12 ] (divided by 2).

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A309138 Nonnegative integers of the form x^2 + 23*y^2.

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