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

franktaw at netscape.net franktaw at netscape.net
Tue Aug 16 20:47:50 CEST 2011

There is a strong tendency for sigma(m) to have more divisors than m
does. I think that explains the rarety of such numbers.

-----Original Message-----
From: Jack Brennen <jfb at brennen.net>

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

<    m, sigma(m), sigma(sigma(m)), ...

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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