Moessner's theorem

In number theory, Moessner's theorem or Moessner's magic is related to an arithmetical algorithm to produce an infinite sequence of the exponents of positive integers ${\displaystyle 1^{n},2^{n},3^{n},4^{n},\cdots ~,}$ with ${\displaystyle n\geq 1~,}$ by recursively manipulating the sequence of integers algebraically. The algorithm was first published by Alfred Moessner in 1951; the first proof of its validity was given by Oskar Perron that same year.

For example, for ${\displaystyle n=2}$, one can remove every even number, resulting in ${\displaystyle (1,3,5,7\cdots )}$, and then add each odd number to the sum of all previous elements, providing ${\displaystyle (1,4,9,16,\cdots )=(1^{2},2^{2},3^{2},4^{2}\cdots )}$.