A Löschian number is an integer of the form
 |
for integers 
and 
. The first few are 0, 1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25,
... (OEIS A003136).
For 
,
the Löschian numbers correspond to the exponents
of the nonzero terms in the Maclaurin series
of
 |
where 
is a q-Pochhammer symbol.
The Löschian numbers are the values 
for Eisenstein integers 
. They are also half the squared Euclidean
norms of vectors in the two-dimensional root
lattice 
,
normalized so that its shortest nonzero vectors have squared
Euclidean norm 2. This is a 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 exponent, 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.
Löschian numbers also occur as curvatures of spheres in a packing related to the bowl of integers (Pegg 2015).
."
9 Aug 2004. 
Pegg, E. Jr. "Loeschian Spheres." Wolfram Demonstrations
Project. 2015.