<s>[INST] even and odd perfect numbers existence Thank for my previous post. Also, thank you so much for this site (m.s.e)
1) If odd perfect numbers there, those numbers can be expressible $12k + 1$ or $324k + 81$
   or $468k + 117$. If yes, please discuss, how far I am correct.
2) If $K$ = $(4^n - 2^n)$/2 is perfect, when $k = 1^3 + 3^3 + ...$
 [/INST] For part 2):
The formula
$$\sum_{k=1}^nk^3=\frac{n^2(n+1)^2}{4}$$
is well-known and leads to 
$$ 1^3+3^3+\ldots+(2m-1)^3=\sum_{k=1}^{2m}k^3-\sum_{k=1}^{m}(2k)^3\\=\frac{(2m)^2(2m+1)^2}{4}-8\cdot\frac{m^2(m+1)^2}{4}\\=m^2(2m^2-1).$$
Since it is well-known that even perfect numbers $N$ are of the form $N=2^{p-1}(2^p-1)$ with $p=2n+1$ an odd prime and $2^p-1$ a Mersenne prime, letting $m=2^n$, you find that indeed $$N=\frac{4^n-2^n}{2}=m^2(2m^2-1)=1^3+3^3+\ldots+(2m-1)^3.$$
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