Loeschian number

From HandWiki
Loeschian number
Named afterAugust Lösch
Subsequence ofA032766
Formulax2 + xy + y2 for integer x, y
First terms0, 1, 3, 4, 7, 9, 12, 13, 16
OEIS index

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).


  • 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).