[seqfan] Sigma-related sequence not in OEIS, need 4th term

Jack Brennen jfb at brennen.net
Mon Aug 15 21:45:30 CEST 2011

a(n) = smallest integer n > 1 such that the first n+1 terms of the 
iterated sigma sequence:

    n, sigma(n), sigma(sigma(n)), ...

all have the same number of divisors.

a(1) = 2
a(2) = 52
a(3) = 4112640

To illustrate:
   n=52; the sequence goes 52, 98, 171, 260, ...

   52, 98, and 171 each have 6 divisors, but 260 has 12 divisors.

What is a(4)?  It exceeds 2.0793*10^10, determined by exhaustive search 
to that limit, but I'm thinking that exhaustive search can probably be 
improved upon through some sort of multi-level sieve...

I find it surprising how rare these numbers become -- numdiv(sigma(X)) 
is a function with a fairly "clumpy" distribution.  And by using 
consecutive starting numbers rather than iterating, we can easily find 
even longer chains, such as the seven numbers from 17331728 to 17331734, 
each of which has numdiv(sigma(X)) equal to 144.  But when iterating, 
duplication seems to be much harder to come by.

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