

A003136


Loeschian numbers: numbers of the form x^2 + xy + y^2; norms of vectors in A2 lattice.
(Formerly M2336)


106



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

1,3


COMMENTS

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).  JeanChristophe 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 unitedged 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
Norms of Eisenstein integers Z[omega] or k(rho).  Juris Evertovskis, Dec 07 2017
Named after the German economist August Lösch (19061945).  Amiram Eldar, Jun 10 2021


REFERENCES

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, SpringerVerlag, 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).


LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000
Joerg Arndt, Planefilling curves on all uniform grids, arXiv preprint arXiv:1607.02433 [math.CO], 2016.
Mira Bernstein, N. J. A. Sloane and Paul E. Wright, On Sublattices of the Hexagonal Lattice, Discrete Math., Vol. 170, No. 13 (1997), pp. 2939; (Abstract, pdf, ps).
John H. Conway, E. M. Rains, and N. J. A. Sloane, On the existence of similar sublattices, Canadian Journal of Mathematics, Vol. 51, No. 6 (1999), pp. 13001306; (Abstract, pdf, ps).
B. N. Delone, Geometry of positive quadratic forms. Part II, Uspekhi Mat. Nauk, Vol. 4 (1938), pp. 102164.
Diane M. Donovan, Terry S. Griggs, Thomas A. McCourt, Jakub Opršal and David Stanovský, Distributive and antidistributive Mendelsohn triple systems, arXiv:1411.5194 [math.CO], (19November2014)
William C. Jagy and Irving Kaplansky, Positive definite binary quadratic forms that represent the same primes [Cached copy]
Yoshiyuki Kitaoka, On the relation between the positivedefinite quadratic forms with the same representation numbers, Proc. Jap. Acad., Vol. 47 (1971), pp. 439441.
August Lösch, Economics of Location, New Haven and London: Yale University Press, 1954. See pp. 117f.
Oscar Marmon, Hexagonal Lattice Points on Circles, arXiv:math/0508201 [math.NT], 2005.
John U. Marshall, The Löschian numbers as a problem in number theory, Geographical Analysis, Vol. 7, No. 4 (1975), pp. 421426; Annotated scanned copy.
John U. Marshall, The construction of the Löschian landscape, Geographical Analysis, Vol. 9, No. 1 (Jan. 1977), pp. 113; Annotated scanned copy.
John U. Marshall, Christallerian networks in the Löschian economic landscape, Professional Geographer, Vol. 29, No. 2 (1977), pp. 153159; Annotated scanned copy.
John U. Marshall, Letter to N. J. A. Sloane, 1990.
Umesh P. Nair, Elementary results on the binary quadratic form a^2+ab+b^2, arXiv:math/0408107 [math.NT], 2004.
Ed Pegg, Jr., Loeschian Spheres, 2015.
Katherine A. Ritchey, Computational Topology for Configuration Spaces of Disks in a Torus, Ph. D. Dissertation, The Ohio State University (2019).
Jacob Rus, Flowsnake Earth, Bridges 2017 Conference Proceedings, pp. 237244.
M. Schering, Théorèmes relatifs aux formes quadratiques qui représentent les mêmes nombres, J. Math, pures el appl., 2 serie 4 (1859).
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references).
V. N. Timofeev, On positive quadratic forms, representing the same numbers, Uspekhi Mat. Nauk, Vol. 18 (1963), pp. 191193.
G. L. Watson, Determination of a binary quadratic form by its values at integer points, Mathematika, Vol. 26, No. 1 (1979), pp. 7275. MR0557128 (81e:10019).
G. L. Watson, Acknowledgement: Determination of a binary quadratic form by its values at integer points, Mathematika, Vol. 27, No. 2 (1980),p. 188. MR0610704 (82d:10037)
Index entries for sequences related to A2 = hexagonal = triangular lattice
Index entries for "core" sequences


FORMULA

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
A088534(a(n)) > 0.  Reinhard Zumkeller, Oct 30 2011
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
From Juris Evertovskis, Dec 07 2017; Jan 01 2020: (Start)
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)


MAPLE

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


MATHEMATICA

ok[n_] := Resolve[Exists[{x, y}, Reduce[n == x^2 + x*y + y^2, {x, y}, Integers]]]; Select[Range[0, 192], ok] (* JeanFrançois 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 (* JeanFrançois Alcover, Nov 27 2015, adapted from PARI *)


PROG

(Haskell)
import Data.Set (singleton, union, fromList, deleteFindMin)
a003136 n = a003136_list !! (n1)
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
 Reinhard Zumkeller, Oct 30 2011
(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
(Magma) [n: n in [0..192]  NormEquation(3, n) eq true]; // Arkadiusz Wesolowski, May 11 2016
(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)]
A003136list(192) > println # Peter Luschny, Mar 17 2018
(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))
A003136_list = list(islice(A003136_gen(), 30)) # Chai Wah Wu, Jan 20 2022


CROSSREFS

Subsequence of A032766, A198772, A118886, A198773, A198774, A198775, A202822, and A260682.
See A092572 for numbers of form x^2 + 3 y^2 with positive x,y.
See A088534 for the number of representations.
Cf. A034020 (complement), A007645 (primes); partitions: A198726, A198727.
Cf. A004611, A034017, A045897, A060428.
Sequence in context: A060428 A352274 A035238 * A326421 A329963 A034022
Adjacent sequences: A003133 A003134 A003135 * A003137 A003138 A003139


KEYWORD

core,easy,nonn,nice


AUTHOR

N. J. A. Sloane


STATUS

approved



