Perfect numbers

Joe Crump joecr at
Wed Apr 18 16:24:57 CEST 2001

Doesn't this just stem from Euler's formula (2^k - 1) * 2^(k-1)?

Obviously the binary rep is going to be a sequence of ones
(the 2^k - 1 term) shifted left k-1 times (the 2^(k-1) term).

The number of ones is prime because k needs to be prime.

- Joe

-----Original Message-----
From: Simon Colton [mailto:simonco at] 
Sent: Wednesday, April 18, 2001 10:14 AM
To: seqfan at

Dear SeqFans,

I thought you might be interested in another cute theorem produced by my
HR program recently.

I asked for supersequences of the perfect numbers which shared at least
hree terms with the perfect numbers. In less than a minute, the program
replied with 47 answers.

The first was:

A052294: pernicious numbers (number of 1s in the binary represenation is

and the last was:

A023578: Nialpdromes (digits in the binary representation are in
descending order)

These two results combined as a conjecture gives us:

The binary representation of every perfect number has a prime number of
1s followed by zeros.

This result can be proved fairly painlessly using Hardy and Wright's
theorem 277. Actually, a stronger version is:

The binary representation of the n-th perfect number consists of the
n-th Mersenne prime (call this k) lots of ones, followed by k-1 zeros.

I like this result because (a) a friend of mine (Jeremy Gow) submitted
pernicious numbers to the Encyclopedia (b) the fact that perfects are
pernicious is appealing and (c) I would never have thought about looking
at the binary representation of perfect numbers.

I imagine that this, or a more general, result is well known, and would
very much appreciate any references.



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