A019280

[Ralf Stephan]

Hello,
the sequence is IMHO irrelevant, as is easily seen. Please comment
if you think this is not so.

ID Number: A019280
Sequence:  1,2,4,6,12,16,18,30,60
Name:   Let sigma_m (n) be result of applying sum-of-divisors function m times
        to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives log_2 of
        the (2,2)-perfect numbers.
References Graeme L. Cohen and Herman J. J. te Riele, Iterating the
              sum-of-divisors function, Experimental Mathematics, 5 (1996), pp.
              93-100.
Links:     Experimental Mathematics, Home Page
See also:  Cf. A019279.
Keywords:  nonn,more
Offset:    1
Author(s): njas
Extension: 2 more terms sent by Jud McCranie (judmccr at bellsouth.net), Jun 01 20
00

Cohen/te Riele show in the reference proofs that
- any even (2,2)-perfect number ("superperfect") must be of the form
  2^(p-1) with 2^p-1 prime (Suryanarayana).
- any odd superperfect number must be a perfect square (Kanold). Searches up 
  to >10^20 did not find any.

Now if there were any superperfects not of the form 2^k, that would break
the sequence since only integers are allowed. OTOH it's already proven that
log_2(even superperfect) = A000043(n)-1 (Mersenne prime exponents).

But all this is contained in the accompanying A019279 (the numbers
themselves), and certainly one would need a new sequence whenever we know 
of odd superperfects. A sequence of log_2(superperfect) however hinges on 
both brittleness and irrelevance.

ralf