Named after August Lösch A032766 x2 + xy + y2 for integer x, y 0, 1, 3, 4, 7, 9, 12, 13, 16 A003136Loeschian 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 ${\displaystyle (m^{2}+m+1)x^{2))$ 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. Retrieved 19 July 2018.