Moessner's theorem

In number theory, Moessner's theorem or Moessner's magic^[1] 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^[2] in 1951; the first proof of its validity was given by Oskar Perron^[3] that same year.^[4]

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 ${\displaystyle (1,4,9,16,\cdots )=(1^{2},2^{2},3^{2},4^{2}\cdots )}$.

Construction

Write down every positive integer and remove every ${\displaystyle n}$-th element, with ${\displaystyle n}$ a positive integer. Build a new sequence of partial sums with the remaining numbers. Continue by removing every ${\displaystyle (n-1)}$-st element in the new sequence and producing a new sequence of partial sums. For the sequence ${\displaystyle k}$, remove the ${\displaystyle (n-k+1)}$-st elements and produce a new sequence of partial sums.

The procedure stops at the ${\displaystyle n}$-th sequence. The remaining sequence will correspond to ${\displaystyle 1^{n},2^{n},3^{n},4^{n}\cdots ~.}$^[4][5]

Example

The initial sequence is the sequence of positive integers,

${\displaystyle 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16\cdots ~.}$

For ${\displaystyle n=4}$, we remove every fourth number from the sequence of integers and add up each element to the sum of the previous elements

${\displaystyle 1,2,3,5,6,7,9,10,11,13,14,15\cdots \to 1,3,6,11,17,24,33,43,54,67,81,96\cdots }$

Now we remove every third element and continue to add up the partial sums

${\displaystyle 1,3,11,17,33,43,67,81\cdots \to 1,4,15,32,65,108,175,256\cdots }$

Remove every second element and continue to add up the partial sums

${\displaystyle 1,15,65,175\cdots \to 1,16,81,256\cdots }$,

which recovers ${\displaystyle 1^{4},2^{4},3^{4},4^{4},\cdots }$.

Variants

If the triangular numbers are removed instead, a similar procedure leads to the sequence of factorials ${\displaystyle 1!,2!,3!,4!,\cdots ~.}$^[1]