Journal of Integer Sequences, Vol. 13 (2010), Article 10.1.5

## Divisibility by 3 of Even Multiperfect Numbers of Abundancy 3 and 4

### Kevin A. Broughan and Qizhi Zhou University of Waikato Hamilton, New Zealand

Abstract:

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 be flat and 4-perfect with exponent and length . If , then is even. If is even and then is also even. If then and is even. If is flat and 3-perfect and , then if , is even. If then is odd. If 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.

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(Concerned with sequences A005820 and A027687.)

Received June 30 2009; revised versions received October 12 2009; January 7 2010. Published in Journal of Integer Sequences, January 8 2010.