Divisibility by 3 of even multiperfect numbers of abundancy 3 and 4
Journal of integer sequences, Tome 13 (2010) no. 1
We say a number is flat if it can be written as a non-trivial power of 2 times an odd squarefree number. The power is the "exponent" and the number of odd primes the "length". Let $N$ be flat and 4-perfect with exponent $a$ and length $m$. If $a\not\equiv 1\bmod 12$, then $a$ is even. If $a$ is even and $3\nmid N$ then $m$ is also even. If $a\equiv 1\bmod 12$ then $3\mid N$ and $m$ is even. If $N$ is flat and 3-perfect and $3\nmid N$, then if $a\not\equiv 1\bmod 12, a$ is even. If $a\equiv 1\bmod 12$ then $m$ is odd. If $N$ is flat and 3 or 4-perfect then it is divisible by at least one Mersenne prime, but not all odd prime divisors are Mersenne. We also give some conditions for the divisibility by 3 of an arbitrary even 4-perfect number.
@article{JIS_2010__13_1_a1,
author = {Broughan, Kevin A. and Zhou, Qizhi},
title = {Divisibility by 3 of even multiperfect numbers of abundancy 3 and 4},
journal = {Journal of integer sequences},
year = {2010},
volume = {13},
number = {1},
zbl = {1206.11008},
language = {en},
url = {http://geodesic.mathdoc.fr/item/JIS_2010__13_1_a1/}
}
Broughan, Kevin A.; Zhou, Qizhi. Divisibility by 3 of even multiperfect numbers of abundancy 3 and 4. Journal of integer sequences, Tome 13 (2010) no. 1. http://geodesic.mathdoc.fr/item/JIS_2010__13_1_a1/