In a posthumously published work, Euler proved that all even
perfect numbers are of the form
![$2^{p-1}(2^p-1)$](abs/img3.svg)
, where
![$2^p-1$](abs/img4.svg)
is
a prime number. In this article, we extend Euler's method for certain
![$\alpha$](abs/img2.svg)
-perfect numbers for which Euler's result can be generalized. In
particular, we use Euler's method to prove that if
![$N$](abs/img5.svg)
is a
![$3$](abs/img6.svg)
-perfect
number divisible by
![$6$](abs/img7.svg)
; then either
![$2\parallel N$](abs/img8.svg)
or
![$3\parallel
N$](abs/img9.svg)
. As well, we prove that if
![$N$](abs/img5.svg)
is a
![$\frac{5}{2}$](abs/img10.svg)
-perfect number
divisible by
![$5$](abs/img11.svg)
, then
![$2^4\parallel N$](abs/img12.svg)
,
![$5^2\parallel N$](abs/img13.svg)
, and
![$31^2\mid
N$](abs/img14.svg)
. Finally, for
![$p\in\{17,257,65537\}$](abs/img15.svg)
, we prove that there are no
![$\frac{2p}{p-1}$](abs/img16.svg)
-perfect numbers divisible by
![$p$](abs/img17.svg)
.
Received July 2 2022; revised version received July 3 2022; October 5 2022; October 6 2022;
October 12 2022.
Published in Journal of Integer Sequences,
October 14 2022.