A084865 -id:A084865 - 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

A002480 Numbers of the form 2x^2 + 3y^2.

Original entry on oeis.org

0, 2, 3, 5, 8, 11, 12, 14, 18, 20, 21, 27, 29, 30, 32, 35, 44, 45, 48, 50, 53, 56, 59, 62, 66, 72, 75, 77, 80, 83, 84, 93, 98, 99, 101, 107, 108, 110, 116, 120, 125, 126, 128, 131, 140, 146, 147, 149, 155, 158, 162

Offset: 1

N. J. A. Sloane, Mira Bernstein

nonn

L. Euler, (E388) Vollstaendige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 425.
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).

Peter Munn, Table of n, a(n) for n = 1..252 (based on A108563 b-file)
L. Euler, Vollstaendige Anleitung zur Algebra, Zweiter Teil.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

For primes see A084865.

Cf. A000075 (growth), A002481, A108563.

A084866 Primes that can be written in the form 2p^2 + 3q^2 with p and q prime.

Original entry on oeis.org

83, 173, 197, 269, 317, 389, 461, 557, 653, 701, 797, 941, 1091, 1109, 1181, 1229, 1637, 1709, 1949, 1997, 2069, 2141, 2309, 2531, 2549, 2621, 2789, 2861, 3221, 3389, 3461, 3581, 3821, 4157, 4229, 4349, 4493, 5051, 5261, 5381, 5501, 5693

Offset: 1

Reinhard Zumkeller, Jun 10 2003

nonn

Subsequence of A084864 and of A084865; A084863(a(n))>0.

			A000040(40) = 173 = 98 + 75 = 2*7^2 + 3*5^2 = 2*A000040(4)^2 + 3*A000040(3)^2, therefore 173 is a term.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A084863, A084864, A084865.

N:= 10^4: # to get terms <= N
P:= select(isprime, [2,seq(i,i=3..floor((N/2)^(1/2)))]):
m:= nops(P):
R:= {}:
for p in P do
  for i from 2 to m while 3*P[i]^2 <= N - 2*p^2 do
    v:= 2*p^2 + 3*P[i]^2;
    if isprime(v) then R:= R union {v} fi
od od:
sort(convert(R,list)); # Robert Israel, Nov 05 2020

nn = 10^4; (* to get terms <= nn *)
P = Select[Join[{2}, Range[3, Floor[Sqrt[nn/2]]]], PrimeQ];
m = Length[P];
R = {};
Do[For[i = 2, 3*P[[i]]^2 <= nn - 2*p^2, i++,
     v = 2*p^2 + 3*P[[i]]^2;
     If[PrimeQ[v], R = R ~Union~ {v}]],
{p, P}];
Sort[R] (* Jean-François Alcover, Dec 13 2021, after Robert Israel *)

A084864 Numbers that can be written in the form 2u^2 + 3v^2, u*v>0.

Original entry on oeis.org

5, 11, 14, 20, 21, 29, 30, 35, 44, 45, 50, 53, 56, 59, 62, 66, 75, 77, 80, 83, 84, 93, 98, 99, 101, 107, 110, 116, 120, 125, 126, 131, 140, 146, 147, 149, 155, 158, 165, 173, 174, 176, 179, 180, 189, 194, 197, 200, 203, 206, 210, 212, 219, 224, 227, 236, 237

Offset: 1

Reinhard Zumkeller, Jun 10 2003

nonn

A084863(a(n)) > 0; for primes see A084865, A084866.

			30 = 18 + 12 = 2*3^2 + 3*2^2, therefore 30 is a term.

Jean-François Alcover, Table of n, a(n) for n = 1..10000

wmax = 240;
Table[w = 2 u^2 + 3 v^2; If[w <= wmax, w, Nothing], {u, 1, Sqrt[wmax/2] // Ceiling}, {v, 1, Sqrt[wmax/3] // Ceiling}] // Flatten // Union (* Jean-François Alcover, Dec 13 2021 *)

A084863 Number of solutions to n = 2u^2 + 3v^2, u*v>0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0

Offset: 1

Reinhard Zumkeller, Jun 10 2003

nonn

a(A084864(n)) > 0.

			n=770 = 2*19^2 + 3*4^2 = 2*17^2 + 3*8^2 = 2*13^2 + 3*12^2 = 2*1 +
3*16^2, therefore a(770)=4.

Eric Weisstein's World of Mathematics, Diophantine Equation: 2nd Powers.

Cf. A084865, A084866.

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

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A002480 Numbers of the form 2x^2 + 3y^2.

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A084866 Primes that can be written in the form 2*p^2 + 3*q^2 with p and q prime.

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A084864 Numbers that can be written in the form 2*u^2 + 3*v^2, u*v>0.

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Mathematica

A084863 Number of solutions to n = 2*u^2 + 3*v^2, u*v>0.

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A084866 Primes that can be written in the form 2p^2 + 3q^2 with p and q prime.

A084864 Numbers that can be written in the form 2u^2 + 3v^2, u*v>0.

A084863 Number of solutions to n = 2u^2 + 3v^2, u*v>0.