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

Benoît Jubin benoit.jubin at gmail.com
Tue Aug 16 11:41:48 CEST 2011

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

There are too many 'n's in here...

Benoit

On Mon, Aug 15, 2011 at 12:45 PM, Jack Brennen <jfb at brennen.net> wrote:
> 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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