for integers and . The first few are 0, 1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25,
... (OEIS A003136).
The Löschian numbers are the norms of Eisenstein integers and can be interpreted as norms of vectors in the two-dimensional root
lattice ,
the triangular lattice (Peterson 1997, p. 53;
Conway and Sloane 1999, p. 111). In the prime
factorization of a positive Löschian number, every prime
number congruent to 2 (mod 3) occurs to an even power, and this condition is
sufficient. Therefore, the product of two Löschian numbers is again Löschian.
The numbers are named after August Lösch, who used them in his theory of economic
location (Lösch 1954, pp. 117-118; Marshall 1975).
Nair (2004) gives elementary results on the corresponding binary quadratic form, and Bhat et al. (2024) use Löschian numbers in a
lozenge triangulation of the plane.
Bhat, R. N.; Cobeli, C.; and Zaharescu, A. "A Lozenge Triangulation of the Plane with Integers." 15 Mar 2024. https://arxiv.org/abs/2403.10500.Conway,
J. H. and Sloane, N. J. A. Sphere
Packings, Lattices and Groups, 3rd ed. New York: Springer-Verlag, p. 111,
1999.Lösch, A. The Economics of Location. New Haven, CT:
Yale University Press, pp. 117-118, 1954.Marshall, J. U. "The
Löschian Numbers as a Problem in Number Theory." Geographical Analysis7,
421-426, 1975.Nair, U. P. "Elementary Results on the Binary
Quadratic Form ."
9 Aug 2004. https://arxiv.org/abs/math/0408107.Pegg,
E. Jr. "Loeschian Spheres." Wolfram Demonstrations Project. 2015.
https://demonstrations.wolfram.com/LoeschianSpheres/.Peterson,
I. The
Jungles of Randomness: A Mathematical Safari. New York: Wiley, p. 53,
1997.Sloane, N. J. A. Sequence A003136/M2336
in "The On-Line Encyclopedia of Integer Sequences."
and 
. The first few are 0, 1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25,
... (OEIS A003136).
,
the triangular lattice (Peterson 1997, p. 53;
Conway and Sloane 1999, p. 111). In the prime
factorization of a positive Löschian number, every prime
number congruent to 2 (mod 3) occurs to an even power, and this condition is
sufficient. Therefore, the product of two Löschian numbers is again Löschian.
The numbers are named after August Lösch, who used them in his theory of economic
location (Lösch 1954, pp. 117-118; Marshall 1975).
."
9 Aug 2004.