Loeschian number

Loeschian number

Named after	August Lösch
Subsequence of	A032766
Formula	x2 + xy + y2 for integer x, y
First terms	0, 1, 3, 4, 7, 9, 12, 13, 16
OEIS index	A003136 Loeschian numbers

In number theory, the numbers of the form x² + xy + y² for integer x, y are called the Loeschian numbers. These numbers are named after August Lösch. They are the norms of the Eisenstein integers. They are a set of whole numbers, including zero, and having prime factorization in which all primes congruent to 2 mod 3 have even powers (there is no restriction of primes congruent to 0 or 1 mod 3).

Properties

• Every Square number is a Loeschian number (by setting x or y to 0).
  • Moreover, every number of the form [math]\displaystyle{ (m^2+m+1)x^2 }[/math] for m and x integers is a Loeschian number (by setting y=mx).

References

• Marshall, J. U. (1975). "The Loeschian numbers as a problem in number theory". Geographical Analysis 7 (4): 421–426. doi:10.1111/j.1538-4632.1975.tb01054.x. 
• "A003136". On-Line Encyclopedia of Integer Sequences. https://oeis.org/A003136.