Geodesic
Divisibility by 3 of even multiperfect numbers of abundancy 3 and 4
Journal of integer sequences, Tome 13 (2010) no. 1.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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.
Classification : 11A05, 11A51
Keywords: multiperfect number, flat number, abundancy 
@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},
     publisher = {mathdoc},
     volume = {13},
     number = {1},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2010__13_1_a1/}
}
TY  - JOUR
AU  - Broughan, Kevin A.
AU  - Zhou, Qizhi
TI  - Divisibility by 3 of even multiperfect numbers of abundancy 3 and 4
JO  - Journal of integer sequences
PY  - 2010
VL  - 13
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JIS_2010__13_1_a1/
LA  - en
ID  - JIS_2010__13_1_a1
ER  - 
%0 Journal Article
%A Broughan, Kevin A.
%A Zhou, Qizhi
%T Divisibility by 3 of even multiperfect numbers of abundancy 3 and 4
%J Journal of integer sequences
%D 2010
%V 13
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JIS_2010__13_1_a1/
%G en
%F 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/