multiperfect number completeness

Don Reble djr at
Tue Nov 4 02:45:10 CET 2003

> From the special form of the even perfects, it should be easy to
> say yea-or-nea to the whole group.  From your evidence below, it
> looks like N>6 -> Nay.

  Ah, of course. The perfect number 2^(p-1)*(2^p-1) is eliminated by the
  sum of the p'th powers of the factors. 6 is the "odd" man in, because
  it's Mersenne exponent is an _even_ prime. Thanks, Rich.

> Given that there are thousands of MPFNs > 6e78, and you've checked the lot-2,
> it looks like your sequence may have come to an end.  Does simply checking
> divisor-sum and sum-of-cubes give a thicker sequence?

  Yes. Checking the same "lot-2", I find 223 numbers in the thicker
  sequence. And all those perfect numbers are in, except 4*(2^3-1), so
  the last two make the total 225.

Don Reble       djr at

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