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A003136 Loeschian numbers: numbers of the form x^2 + xy + y^2; norms of vectors in A2 lattice.
(Formerly M2336)
118

%I M2336 #289 Mar 18 2024 07:43:00

%S 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,

%T 64,67,73,75,76,79,81,84,91,93,97,100,103,108,109,111,112,117,121,124,

%U 127,129,133,139,144,147,148,151,156,157,163,169,171,172,175,181,183,189,192

%N Loeschian numbers: numbers of the form x^2 + xy + y^2; norms of vectors in A2 lattice.

%C Equally, numbers of the form x^2 - xy + y^2. - _Ray Chandler_, Jan 27 2009

%C Also, numbers of the form X^2+3Y^2 (X=y+x/2, Y=x/2), cf. A092572. - _Zak Seidov_, Jan 20 2009

%C 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

%C Equivalently, numbers n such that the coefficient of x^n in Theta3(x)*Theta3(x^3) is nonzero. - _Joerg Arndt_, Jan 16 2011

%C 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

%C Relative areas of equilateral triangles whose vertices are on a triangular lattice. - Anton Sherwood (bronto(AT)pobox.com), Apr 05 2001

%C 2 appended to a(n) (for positive n) corresponds to capsomere count in viral architectural structures (cf. A071336). - _Lekraj Beedassy_, Apr 14 2006

%C The triangle in A132111 gives the enumeration: n^2 + k*n + k^2, 0 <= k <= n.

%C The number of coat proteins at each corner of a triangular face of a virus shell. - _Parthasarathy Nambi_, Sep 04 2007

%C 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

%C 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

%C 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

%C Equivalently, the existence spectrum of affine Mendelsohn triple systems, cf. A248107. - _David Stanovsky_, Nov 25 2014

%C 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

%C 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

%C Norms of Eisenstein integers Z[omega] or k(rho). - _Juris Evertovskis_, Dec 07 2017

%C Named after the German economist August Lösch (1906-1945). - _Amiram Eldar_, Jun 10 2021

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

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

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A003136/b003136.txt">Table of n, a(n) for n = 1..10000</a>

%H Joerg Arndt, <a href="https://arxiv.org/abs/1607.02433">Plane-filling curves on all uniform grids</a>, arXiv preprint arXiv:1607.02433 [math.CO], 2016.

%H Mira Bernstein, N. J. A. Sloane and Paul E. Wright, <a href="https://doi.org/10.1016/0012-365X(95)00354-Y">On Sublattices of the Hexagonal Lattice</a>, Discrete Math., Vol. 170, No. 1-3 (1997), pp. 29-39; (<a href="http://neilsloane.com/doc/paul.txt">Abstract</a>, <a href="http://neilsloane.com/doc/paul.pdf">pdf</a>, <a href="http://neilsloane.com/doc/paul.ps">ps</a>).

%H Raghavendra N. Bhat, Cristian Cobeli, and Alexandru Zaharescu, <a href="https://arxiv.org/abs/2403.10500">A lozenge triangulation of the plane with integers</a>, arXiv:2403.10500 [math.NT], 2024.

%H John H. Conway, E. M. Rains, and N. J. A. Sloane, <a href="https://doi.org/10.4153/CJM-1999-059-5">On the existence of similar sublattices</a>, Canadian Journal of Mathematics, Vol. 51, No. 6 (1999), pp. 1300-1306; (<a href="http://neilsloane.com/doc/sim.txt">Abstract</a>, <a href="http://neilsloane.com/doc/sim.pdf">pdf</a>, <a href="http://neilsloane.com/doc/sim.ps">ps</a>).

%H B. N. Delone, <a href="http://mi.mathnet.ru/eng/umn7118">Geometry of positive quadratic forms. Part II</a>, Uspekhi Mat. Nauk, Vol. 4 (1938), pp. 102-164.

%H Diane M. Donovan, Terry S. Griggs, Thomas A. McCourt, Jakub Opršal and David Stanovský, <a href="http://arxiv.org/abs/1411.5194">Distributive and anti-distributive Mendelsohn triple systems</a>, arXiv:1411.5194 [math.CO], (19-November-2014)

%H William C. Jagy and Irving Kaplansky, <a href="/A244019/a244019.pdf">Positive definite binary quadratic forms that represent the same primes</a> [Cached copy]

%H Yoshiyuki Kitaoka, <a href="http://dx.doi.org/10.3792/pja/1195519924">On the relation between the positive-definite quadratic forms with the same representation numbers</a>, Proc. Jap. Acad., Vol. 47 (1971), pp. 439-441.

%H August Lösch, <a href="http://archive.org/stream/economicsoflocat00ls#page/116/mode/2up">Economics of Location</a>, New Haven and London: Yale University Press, 1954. See pp. 117f.

%H Oscar Marmon, <a href="http://arxiv.org/abs/math/0508201">Hexagonal Lattice Points on Circles</a>, arXiv:math/0508201 [math.NT], 2005.

%H John U. Marshall, <a href="https://doi.org/10.1111/j.1538-4632.1975.tb01054.x">The Löschian numbers as a problem in number theory</a>, Geographical Analysis, Vol. 7, No. 4 (1975), pp. 421-426; <a href="/A003136/a003136_2.pdf">Annotated scanned copy</a>.

%H John U. Marshall, <a href="https://doi.org/10.1111/j.1538-4632.1977.tb00555.x">The construction of the Löschian landscape</a>, Geographical Analysis, Vol. 9, No. 1 (Jan. 1977), pp. 1-13; <a href="/A003136/a003136_3.pdf">Annotated scanned copy</a>.

%H John U. Marshall, <a href="https://doi.org/10.1111/j.0033-0124.1977.00153.x">Christallerian networks in the Löschian economic landscape</a>, Professional Geographer, Vol. 29, No. 2 (1977), pp. 153-159; <a href="/A003136/a003136_1.pdf">Annotated scanned copy</a>.

%H John U. Marshall, <a href="/A003136/a003136.pdf">Letter to N. J. A. Sloane, 1990</a>.

%H Umesh P. Nair, <a href="http://arXiv.org/abs/math.NT/0408107">Elementary results on the binary quadratic form a^2+ab+b^2</a>, arXiv:math/0408107 [math.NT], 2004.

%H Ed Pegg, Jr., <a href="http://demonstrations.wolfram.com/LoeschianSpheres/">Loeschian Spheres</a>, 2015.

%H Olivier Ramaré, S. Ettahri, and L. Surel, <a href="https://hal.science/hal-03381427">Fast multi-precision computation of some Euler products</a>, Mathematics of Computation (2021) hal-03381427.

%H Katherine A. Ritchey, <a href="http://rave.ohiolink.edu/etdc/view?acc_num=osu1562945889197152">Computational Topology for Configuration Spaces of Disks in a Torus</a>, Ph. D. Dissertation, The Ohio State University (2019).

%H Jacob Rus, <a href="https://archive.bridgesmathart.org/2017/bridges2017-237.pdf">Flowsnake Earth</a>, Bridges 2017 Conference Proceedings, pp. 237-244.

%H M. Schering, <a href="http://sites.mathdoc.fr/JMPA/PDF/JMPA_1859_2_4_A24_0.pdf">Théorèmes relatifs aux formes quadratiques qui représentent les mêmes nombres</a>, J. Math, pures el appl., 2 serie 4 (1859).

%H N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references).

%H James Smith, <a href="https://arxiv.org/abs/2403.01911">Turtles, Hats and Spectres: Aperiodic structures on a Rhombic tiling</a>, arXiv:2403.01911 [math.MG], 2024. See p. 18.

%H V. N. Timofeev, <a href="http://mi.mathnet.ru/eng/umn6393">On positive quadratic forms, representing the same numbers</a>, Uspekhi Mat. Nauk, Vol. 18 (1963), pp. 191-193.

%H G. L. Watson, <a href="http://dx.doi.org/10.1112/S0025579300009621">Determination of a binary quadratic form by its values at integer points</a>, Mathematika, Vol. 26, No. 1 (1979), pp. 72-75. MR0557128 (81e:10019).

%H G. L. Watson, <a href="http://dx.doi.org/10.1112/S002557930001007X">Acknowledgement: Determination of a binary quadratic form by its values at integer points</a>, Mathematika, Vol. 27, No. 2 (1980),p. 188. MR0610704 (82d:10037)

%H <a href="/index/Aa#A2">Index entries for sequences related to A2 = hexagonal = triangular lattice</a>

%H <a href="/index/Cor#core">Index entries for "core" sequences</a>

%F 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).

%F 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

%F A088534(a(n)) > 0. - _Reinhard Zumkeller_, Oct 30 2011

%F 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

%F Comments from Richard C. Schroeppel, Jul 20 2016: (Start)

%F The set is also closed under restricted division: If M and N are members, and M divides N, then N/M is a member.

%F If N == 2 (mod 3), N is not in the sequence.

%F 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)).

%F Representations usually come in sets of 6: (K,L), (K+L,-K), (K+L,-L) and their negatives. (End)

%F 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

%F From _Juris Evertovskis_, Dec 07 2017; Jan 01 2020: (Start)

%F 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.

%F 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)

%p 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

%t 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 *)

%t 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 *)

%t 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 *)

%o (Haskell)

%o import Data.Set (singleton, union, fromList, deleteFindMin)

%o a003136 n = a003136_list !! (n-1)

%o a003136_list = f 0 $ singleton 0 where

%o f x s | m < x ^ 2 = m : f x s'

%o | otherwise = m : f x'

%o (union s' $ fromList $ map (\y -> x'^2+(x'+y)*y) [0..x'])

%o where x' = x + 1

%o (m,s') = deleteFindMin s

%o -- _Reinhard Zumkeller_, Oct 30 2011

%o (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)

%o (PARI) is(n)=#bnfisintnorm(bnfinit(z^2+z+1),n) \\ _Ralf Stephan_, Oct 18 2013

%o (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

%o (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

%o (PARI) is_a003136(n) = !n || #qfbsolve(Qfb(1, 1, 1), n, 3) \\ _Hugo Pfoertner_, Aug 04 2023

%o (Magma) [n: n in [0..192] | NormEquation(3, n) eq true]; // _Arkadiusz Wesolowski_, May 11 2016

%o (Julia)

%o function isA003136(n)

%o n % 3 == 2 && return false

%o n in [0, 1, 3] && return true

%o M = Int(round(2*sqrt(n/3)))

%o for y in 0:M, x in 0:y

%o n == x^2 + y^2 + x*y && return true

%o end

%o return false

%o end

%o A003136list(upto) = [n for n in 0:upto if isA003136(n)]

%o A003136list(192) |> println # _Peter Luschny_, Mar 17 2018

%o (Python)

%o from itertools import count, islice

%o from sympy import factorint

%o 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))

%o A003136_list = list(islice(A003136_gen(),30)) # _Chai Wah Wu_, Jan 20 2022

%Y Subsequence of A032766, A198772, A118886, A198773, A198774, A198775, A202822, and A260682.

%Y See A092572 for numbers of form x^2 + 3 y^2 with positive x,y.

%Y See A088534 for the number of representations.

%Y Cf. A034020 (complement), A007645 (primes); partitions: A198726, A198727.

%Y Cf. A004611, A034017, A045897, A060428.

%K core,easy,nonn,nice,changed

%O 1,3

%A _N. J. A. Sloane_

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)