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A003136
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Loeschian numbers: numbers of the form x^2 + xy + y^2; norms of vectors in A2 lattice.
(Formerly M2336)
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118
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0, 1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25, 27, 28, 31, 36, 37, 39, 43, 48, 49, 52, 57, 61, 63, 64, 67, 73, 75, 76, 79, 81, 84, 91, 93, 97, 100, 103, 108, 109, 111, 112, 117, 121, 124, 127, 129, 133, 139, 144, 147, 148, 151, 156, 157, 163, 169, 171, 172, 175, 181, 183, 189, 192
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OFFSET
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1,3
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COMMENTS
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Equally, numbers of the form x^2 - xy + y^2. - Ray Chandler, Jan 27 2009
Also, numbers of the form X^2+3Y^2 (X=y+x/2, Y=x/2), cf. A092572. - Zak Seidov, Jan 20 2009
Theorem (Schering, Delone, Watson): The only positive definite binary quadratic forms that represent the same numbers are x^2+xy+y^2 and x^2+3y^2 (up to scaling). - N. J. A. Sloane, Jun 22 2014
Equivalently, numbers n such that the coefficient of x^n in Theta3(x)*Theta3(x^3) is nonzero. - Joerg Arndt, Jan 16 2011
Equivalently, numbers n such that the coefficient of x^n in a(x) (resp. b(x)) is nonzero where a(), b() are cubic AGM functions. - Michael Somos, Jan 16 2011
Relative areas of equilateral triangles whose vertices are on a triangular lattice. - Anton Sherwood (bronto(AT)pobox.com), Apr 05 2001
2 appended to a(n) (for positive n) corresponds to capsomere count in viral architectural structures (cf. A071336). - Lekraj Beedassy, Apr 14 2006
The triangle in A132111 gives the enumeration: n^2 + k*n + k^2, 0 <= k <= n.
The number of coat proteins at each corner of a triangular face of a virus shell. - Parthasarathy Nambi, Sep 04 2007
Numbers of the form (x^2 + y^2 + (x + y)^2)/2. If we let z = - x - y, then all the solutions to x^2 + y^2 + z^2 = k with x + y + z = 0 are k = 2a(n) for any n. - Jon Perry, Dec 16 2012
Sequence of divisors of the hexagonal lattice, except zero (where it is said that an integer n divides a lattice if there exists a sublattice of index n; example: 3 divides the hexagonal lattice). - Jean-Christophe Hervé, May 01 2013
Numbers of the form - (x*y + y*z + x*z) with x + y + z = 0. Numbers of the form x^2 + y^2 + z^2 - (x*y + y*z + x*z) = (x - y)*(x - z) + (y - x) * (y - z) + (z - x) * (z - y). - Michael Somos, Jun 26 2013
Equivalently, the existence spectrum of affine Mendelsohn triple systems, cf. A248107. - David Stanovsky, Nov 25 2014
Lame's solutions to the Helmholtz equation with Dirichlet boundary conditions on the unit-edged equilateral triangle have eigenvalues of the form: (x^2+x*y+y^2)*(4*Pi/3)^2. The actual set, starting at 1 and counting degeneracies, is given by A060428, e.g., the first degeneracy is 49 where (x,y)=(0,7) and (3,5). - Robert Stephen Jones, Oct 01 2015
Curvatures of spheres in one bowl of integers, the Loeschian spheres. Mod 12, numbers equal to 0, 1, 3, 4, 7, 9. - Ed Pegg Jr, Jan 10 2017
Named after the German economist August Lösch (1906-1945). - Amiram Eldar, Jun 10 2021
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REFERENCES
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J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 111.
Ivars Peterson, The Jungles of Randomness: A Mathematical Safari, John Wiley and Sons, (1998) pp. 53.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Jacob Rus, Flowsnake Earth, Bridges 2017 Conference Proceedings, pp. 237-244.
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FORMULA
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Either n=0 or else in the prime factorization of n all primes of the form 3a+2 must occur to even powers only (there is no restriction of primes congruent to 0 or 1 mod 3).
If n is in the sequence, then n^k is in the sequence (but the converse is not true). n is in the sequence iff n^(2k+1) is in the sequence. - Ray Chandler, Feb 03 2009
The sequence is multiplicative in the sense that if m and n are in the sequence, so is m*n. - Jon Perry, Dec 18 2012
Comments from Richard C. Schroeppel, Jul 20 2016: (Start)
The set is also closed under restricted division: If M and N are members, and M divides N, then N/M is a member.
If N == 2 (mod 3), N is not in the sequence.
The density of members (relative to the integers>0) gradually falls to 0. The density goes as O(1/sqrt(log N)). This implies that, if N is a member, the average expected number of representations of N is O(sqrt(log N)).
Representations usually come in sets of 6: (K,L), (K+L,-K), (K+L,-L) and their negatives. (End)
Since Q(zeta), where zeta is a primitive 3rd root of unity has class number 1, the situation as to whether an integer is of the form x^2 + xy + y^2 is similar to the situation with x^2 + y^2: n is of the that form if and only if every prime p dividing n which is = 5 mod 6 divides it to an even power. The density of 1/sqrt(x) that Rich mentioned is an old result due to Landau. - Victor S. Miller, Jul 20 2016
In the prime factorization of n, let S_1 be the set of distinct prime factors p_i for which p_i == 1 (mod 3), let S_2 be the set of distinct prime factors p_j for which p_j == 2 (mod 3), and let M be the exponent of 3. Then n = 3^M * (Product_{p_i in S_1} p_i ^ e_i) * (Product_{p_j in S_2} p_j ^ e_j), and the number of solutions for x^2+xy+y^2=n is 6*Product_{p_i in S_1} (e_i + 1) if all e_j are even and 0 otherwise.
For all Löschian numbers there are nonnegative X,Y such that X^2+XY+Y^2=n. For x,y such that x^2+xy+y^2=n take X=min(|x|,|y|), Y=|x+y| if xy<0 and X=|x|, Y=|y| otherwise. (End)
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MAPLE
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readlib(ifactors): for n from 2 to 200 do m := ifactors(n)[2]: flag := 1: for i from 1 to nops(m) do if m[i, 1] mod 3 = 2 and m[i, 2] mod 2 = 1 then flag := 0; break fi: od: if flag=1 then printf(`%d, `, n) fi: od: # James A. Sellers, Dec 07 2000
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MATHEMATICA
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ok[n_] := Resolve[Exists[{x, y}, Reduce[n == x^2 + x*y + y^2, {x, y}, Integers]]]; Select[Range[0, 192], ok] (* Jean-François Alcover, Apr 18 2011 *)
nn = 14; Select[Union[Flatten[Table[x^2 + x*y + y^2, {x, 0, nn}, {y, 0, x}]]], # <= nn^2 &] (* T. D. Noe, Apr 18 2011 *)
QP = QPochhammer; s = QP[q]^3 / QP[q^3]/3 + O[q]^200; Position[ CoefficientList[s, q], n_ /; n != 0] - 1 // Flatten (* Jean-François Alcover, Nov 27 2015, adapted from PARI *)
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PROG
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(Haskell)
import Data.Set (singleton, union, fromList, deleteFindMin)
a003136 n = a003136_list !! (n-1)
a003136_list = f 0 $ singleton 0 where
f x s | m < x ^ 2 = m : f x s'
| otherwise = m : f x'
(union s' $ fromList $ map (\y -> x'^2+(x'+y)*y) [0..x'])
where x' = x + 1
(m, s') = deleteFindMin s
(PARI) isA003136(n)=local(fac, flag); if(n==0, 1, fac=factor(n); flag=1; for(i=1, matsize(fac)[1], if(Mod(fac[i, 1], 3)==2 && Mod(fac[i, 2], 2)==1, flag=0)); flag)
(PARI) is(n)=#bnfisintnorm(bnfinit(z^2+z+1), n) \\ Ralf Stephan, Oct 18 2013
(PARI) x='x+O('x^200); p=eta(x)^3/eta(x^3); for(n=0, 199, if(polcoeff(p, n) != 0, print1(n, ", "))) \\ Altug Alkan, Nov 08 2015
(PARI) list(lim)=my(v=List(), y, t); for(x=0, sqrtint(lim\3), my(y=x, t); while((t=x^2+x*y+y^2)<=lim, listput(v, t); y++)); Set(v) \\ Charles R Greathouse IV, Feb 05 2017
(PARI) is_a003136(n) = !n || #qfbsolve(Qfb(1, 1, 1), n, 3) \\ Hugo Pfoertner, Aug 04 2023
(Julia)
function isA003136(n)
n % 3 == 2 && return false
n in [0, 1, 3] && return true
M = Int(round(2*sqrt(n/3)))
for y in 0:M, x in 0:y
n == x^2 + y^2 + x*y && return true
end
return false
end
A003136list(upto) = [n for n in 0:upto if isA003136(n)]
(Python)
from itertools import count, islice
from sympy import factorint
def A003136_gen(): return (n for n in count(0) if all(e % 2 == 0 for p, e in factorint(n).items() if p % 3 == 2))
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CROSSREFS
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See A092572 for numbers of form x^2 + 3 y^2 with positive x,y.
See A088534 for the number of representations.
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KEYWORD
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core,easy,nonn,nice
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AUTHOR
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STATUS
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approved
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