A003136 Loeschian numbers: numbers of the form x^2 + xy + y^2; norms of vectors in A2 lattice.
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
Offset: 1
References
- 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).
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.
- Shreeya Behera, Variants of the chromatic number of the plane, Ph. D. dissertation, Ohio State Univ., 2024. See p. 88.
- Mira Bernstein, N. J. A. Sloane and Paul E. Wright, On Sublattices of the Hexagonal Lattice, Discrete Math., Vol. 170, No. 1-3 (1997), pp. 29-39; (Abstract, pdf, ps).
- Raghavendra N. Bhat, Cristian Cobeli, and Alexandru Zaharescu, A lozenge triangulation of the plane with integers, arXiv:2403.10500 [math.NT], 2024.
- 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. 1300-1306; (Abstract, pdf, ps).
- B. N. Delone, Geometry of positive quadratic forms. Part II, Uspekhi Mat. Nauk, Vol. 4 (1938), pp. 102-164.
- Diane M. Donovan, Terry S. Griggs, Thomas A. McCourt, Jakub Opršal and David Stanovský, Distributive and anti-distributive Mendelsohn triple systems, arXiv:1411.5194 [math.CO], (19-November-2014).
- Shyam Sunder Gupta, On Some Special Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 22, 527-565.
- A. Itin, New geometric receipts for design of photonic crystals and metamaterials: optimal toric packings, arXiv:2410.17424 [physics.app-ph], 2024. See p. 4.
- 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 positive-definite quadratic forms with the same representation numbers, Proc. Jap. Acad., Vol. 47 (1971), pp. 439-441.
- 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. 421-426; Annotated scanned copy.
- John U. Marshall, The construction of the Löschian landscape, Geographical Analysis, Vol. 9, No. 1 (Jan. 1977), pp. 1-13; Annotated scanned copy.
- John U. Marshall, Christallerian networks in the Löschian economic landscape, Professional Geographer, Vol. 29, No. 2 (1977), pp. 153-159; 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.
- Olivier Ramaré, S. Ettahri, and L. Surel, Fast multi-precision computation of some Euler products, Mathematics of Computation (2021) hal-03381427.
- 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. 237-244.
- 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).
- James Smith, Turtles, Hats and Spectres: Aperiodic structures on a Rhombic tiling, arXiv:2403.01911 [math.MG], 2024. See p. 18.
- V. N. Timofeev, On positive quadratic forms, representing the same numbers, Uspekhi Mat. Nauk, Vol. 18 (1963), pp. 191-193.
- G. L. Watson, Determination of a binary quadratic form by its values at integer points, Mathematika, Vol. 26, No. 1 (1979), pp. 72-75. 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
Crossrefs
Programs
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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 -- Reinhard Zumkeller, Oct 30 2011
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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
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Magma
[n: n in [0..192] | NormEquation(3, n) eq true]; // Arkadiusz Wesolowski, May 11 2016
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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 Sellers, Dec 07 2000
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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 *) -
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)
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PARI
is(n)=#bnfisintnorm(bnfinit(z^2+z+1),n) \\ Ralf Stephan, Oct 18 2013
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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
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PARI
is_a003136(n) = !n || #qfbsolve(Qfb(1, 1, 1), n, 3) \\ Hugo Pfoertner, Aug 04 2023
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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
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 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)
Comments