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A003136 Loeschian numbers: numbers of the form x^2 + xy + y^2; norms of vectors in A2 lattice.
(Formerly M2336)
91
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 (list; graph; refs; listen; history; text; internal format)
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

A088534(a(n)) > 0. - Reinhard Zumkeller, Oct 30 2011

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

The number of structure units in an icosahedral virus is 20*a(n) for an integer n, see Stannard link. - Charles R Greathouse IV, Nov 03 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

REFERENCES

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111.

J. U. Marshall, The Loeschian numbers as a problem in number theory, Geographical Analysis, 7 (1975), 421-426.

J. U. Marshall, The construction of the Loschian landscape, Geographical Analysis, 9 (Jan. 1977), 1-13.

J. U. Marshall, Christallerian networks in the Loschian economic landscape, Professional Geographer, 29 (No. 2, 1977), 153-159.

Ivars Peterson, "The Jungles of Randomness: A Mathematical Safari", John Wiley and Sons, (1998) pp. 53.

M. Schering. Theoremes relatifs aux formes quadratiques qui representent les memes nombres, J. Math, pures el appl., 2 serie 4 (1859).

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, Plane-filling curves on all uniform grids, arXiv preprint arXiv:1607.02433 [math.CO], 2016.

M. Bernstein, N. J. A. Sloane and P. E. Wright, On Sublattices of the Hexagonal Lattice, Discrete Math. 170 (1997) 29-39 (Abstract, pdf, ps).

J. H. Conway, E. M. Rains and N. J. A. Sloane, On the existence of similar sublattices, Canad. J. Math. 51 (1999), 1300-1306 (Abstract, pdf, ps).

B. N. Delone, Geometry of positive quadratic forms. Part II, Uspekhi Mat. Nauk, 4 (1938), 102-164.

Diane M. Donovan, Terry S. Griggs, Thomas A. McCourt, Jakub Opršal, David Stanovský, Distributive and anti-distributive Mendelsohn triple systems, arXiv:1411.5194 [math.CO], (19-November-2014)

William C. Jagy and Irving Kaplansky, Positive definite binary quadratic forms that represent the same primes [Cached copy]

Y. Kitaoka, On the relation between the positive-definite quadratic forms with the same representation numbers, Proc. Jap. Acad., 47 (1971), 439-441.

August Lösch, Economics of Location (1954), see pp. 117f.

Oscar Marmon, Hexagonal Lattice Points on Circles, arXiv:math/0508201 [math.NT], 2005.

J. U. Marshall, Christallerian networks in the Loschian economic landscape, Professional Geographer, 29 (No. 2, 1977), 153-159. [Annotated scanned copy]

J. U. Marshall, The construction of the Loschian landscape, Geographical Analysis, 9 (Jan. 1977), 1-13. [Annotated scanned copy]

J. U. Marshall, Letter to N. J. A. Sloane, 1990.

J. U. Marshall, The Loeschian numbers as a problem in number theory, Geographical Analysis, 7 (1975), 421-426. [Annotated scanned copy]

U. P. Nair, Elementary results on the binary quadratic form a^2+ab+b^2, arXiv:math/0408107 [math.NT], 2004.

E. Pegg Jr, Loeschian Spheres, 2015.

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

Linda Stannard, Principles of Virus Architecture (archived version)

V. N. Timofeev, On positive quadratic forms, representing the same numbers, Uspekhi Mat. Nauk, 18 (1963), 191-193.

G. L. Watson, Determination of a binary quadratic form by its values at integer points, Mathematika 26 (1979), no. 1, 72--75. MR0557128 (81e:10019).

G. L. Watson, Acknowledgement: Determination of a binary quadratic form by its values at integer points, Mathematika 27 (1980), no. 2, 188 (1981). 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

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: (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. Take X=|x|, Y=|x+y| for any x,y such that x^2+xy+y^2=n. (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] (* 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 *)

PROG

(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

-- 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

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: A120451 A060428 A035238 * A034022 A198772 A185256

Adjacent sequences:  A003133 A003134 A003135 * A003137 A003138 A003139

KEYWORD

core,easy,nonn,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 18 15:07 EST 2018. Contains 318229 sequences. (Running on oeis4.)