cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

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

Keywords

Comments

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.

References

  • 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.

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)
  • PARI
    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

Extensions

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

A020674 Numbers of the form 2*x^2 + 5*y^2.

Original entry on oeis.org

0, 2, 5, 7, 8, 13, 18, 20, 22, 23, 28, 32, 37, 38, 45, 47, 50, 52, 53, 55, 63, 70, 72, 77, 80, 82, 88, 92, 95, 98, 103, 112, 117, 118, 125, 127, 128, 130, 133, 143, 148, 152, 157, 162, 167, 173, 175, 178, 180, 182, 188, 197, 198, 200, 205, 207, 208, 212, 220, 223, 230, 242
Offset: 1

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Cf. A000286, A106889 (primes).

Programs

  • Mathematica
    Union[2*First[#]+5*Last[#]&/@(Tuples[Range[0,10],2]^2)] (* Harvey P. Dale, May 09 2012 *)

A317642 Expansion of theta_3(q^2)*theta_3(q^5), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 0, 2, 0, 0, 2, 0, 4, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, 0, 2, 0, 4, 4, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 2, 0, 4, 0, 0, 2, 0, 4, 4, 0, 4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 0, 8, 0, 0, 2, 0, 4, 0, 0, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 4, 0, 0, 6
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 02 2018

Keywords

Comments

Number of integer solutions to the equation 2*x^2 + 5*y^2 = n.

Examples

			G.f. = 1 + 2*q^2 + 2*q^5 + 4*q^7 + 2*q^8 + 4*q^13 + 2*q^18 + 2*q^20 + 4*q^22 + ...

Links

Crossrefs

Cf. A000286, A020674, A033718, A106889, A108563, A192323.

Programs

  • Mathematica
    nmax = 98; CoefficientList[Series[EllipticTheta[3, 0, q^2] EllipticTheta[3, 0, q^5], {q, 0, nmax}], q]
nmax = 98; CoefficientList[Series[QPochhammer[-q^2, -q^2] QPochhammer[-q^5, -q^5]/(QPochhammer[q^2, -q^2] QPochhammer[q^5, -q^5]), {q, 0, nmax}], q]

Formula

G.f.: Product_{k>=1} (1 + x^(4*k-2))^2*(1 - x^(4*k))*(1 + x^(10*k-5))^2*(1 - x^(10*k)).
Showing 1-3 of 3 results.

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