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Löschian number

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Löschian 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 Löschian numbers (or 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

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  • Every Löschian number is nonnegative.
  • Every square number is a Löschian number (by setting x or y to 0).
    • Moreover, every number of the form for m and x integers is a Löschian number (by setting y=mx).
  • There are infinitely many Löschian numbers.
  • Given that odd and even integers are equally numerous, the probability that a Löschian number is odd is 0.75, and the probability that it is even is 0.25. This follows from the fact that is even only if x and y are both even.
  • The greatest common divisor and the least common multiple of any two or more Löschian numbers are also Löschian numbers.
  • The product of two Löschian numbers is always a Löschian number, in other words Löschian numbers are closed under multiplication.
    • This property makes the set of Löschian numbers into a monoid under multiplication.
  • The product of a Löschian number and a non-Löschian number is never a Löschian number.

References

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  • 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.