[seqfan] Re: Do 20 and 105 eventually reach a prime in these sequences?

[Charles Greathouse]

> Most *small* integers quickly reach a prime.  I'm confident that most
> integers do not.

Interesting.  The numbers don't grow that quickly, so I'd think you'd
typically reach a prime just 'by chance'.  I guess it all depends on
their rate of increase: the harmonic series diverges but the sum of
1/log(n log n) converges.

In terms of factors out front it looks favorable: it should be very
rare to have the numbers divisible by 2 or 5.  Actually, the latter
suggests a sequence 4, 9, 16, 18, 25, 36, 49, ... "Numbers n for which
the exponent of the largest prime factor of n is even.".  Worth
submitting?

McNeil: I was wondering who beat me to factordb!

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Tue, Sep 13, 2011 at 4:38 PM,  <franktaw at netscape.net> wrote:
> Most *small* integers quickly reach a prime.  I'm confident that most
> integers do not.
>
> Franklin T. Adams-Watters
>
> -----Original Message-----
> From: Alonso Del Arte <alonso.delarte at gmail.com>
>
> ...
> If we take the factorization of an integer, say 18 = 2 * 3^2, strip out the
> operators and concatenate the integers, then repeat the operation, most
> integers quickly reach a prime. Indeed, 232 = 2^3 * 29, and 2329 = 17 * 137,
> and 17137 is prime.
>
> ...
>
> Do 20 and 105 reach a prime under this process? With 20 we get 20, 225,
> 3252, 223271, 297699, 399233, 715623, 3263907, 32347303, ... I've taken this
> to forty steps, and 4339779194757514315803243245042123341102411963 is
> composite. Similary with 105: 105, 357, 3717, 32759, 174147, 358049, 379677,
> 3196661, 13245897, ... and at forty steps,
> 7903796151050019316957414263672508157280737 is composite.
>
>
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