Has this conjecture been proven/disproven/found before? My 11-year-old son has made the following conjecture, and wants to know if it's already known to be true or false:
if n is a whole number,
    define an operation #(n) = 1+2+3…..+n (like factorial but with addition)
    The conjecture: if n is odd, then #(n-1) = X*n (where X may be any natural number). That is, #(n-1) should be evenly divisible by n.

$2\#(n)=(1+2+\cdots+n)+(n+n-1+\cdots+1)=(1+n)+(2+n-1)+\cdots+(n+1)=(n+1)+(n+1)+\cdots+(n+1)=n(n+1)$
So $\#(n)=\frac{1}{2}n(n+1)$.
Now $\#(n-1)=\frac{1}{2}(n-1)n$
This is divible by $n$ if $\frac{1}{2}(n-1)$ is an integer.
If $n$ is odd then $n-1$ is even so $\frac{1}{2}(n-1)$ is indeed an integer, therefore the conjecture is true!
