[seqfan] Re: possible sequence

franktaw at netscape.net franktaw at netscape.net
Fri Mar 7 16:48:18 CET 2014


If 2^86 is zero-free, with 26 digits, then the 26 digit cycle contains 
zero-free numbers. So this would have to be at least the 27-digit cycle.

Franklin T. Adams-Watters

-----Original Message-----
From: Allan Wechsler <acwacw at gmail.com>
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Sent: Fri, Mar 7, 2014 8:35 am
Subject: [seqfan] Re: possible sequence


I can vaguely imagine proving an upper bound on A007377, since any fixed
number of low-order digits of the powers of two must eventually enter a
cycle.  If one could actually step through, say, the 15-digit cycle, and
find that every step had a zero ...


On Thu, Mar 6, 2014 at 9:06 PM, Charles Greathouse <
charles.greathouse at case.edu> wrote:

> 2^86 (26 digits) is conjectured to be the last 0-free power of two,
> see A007377.
>
> 2^184 * 3^88 (98 digits) seems to be the last 0-free 3-smooth number.
>
> 0-free numbers cannot contain both 2s and 5s, and the best {3, 
5}-smooth
> number I can find is 3^28 * 5^90 (77 digits), so it looks like the 
largest
> 0-free 5-smooth number is 2^184 * 3^88 again.
>
> I don't know what the largest 0-free 7-smooth number is, but it's at
> least 2^298 * 3^69 * 7^7 (129 digits).
>
> One problem is that this sequence grows very quickly. Another is that 
no
> terms are actually known...! But it is interesting to think about.
>
> Charles Greathouse
> Analyst/Programmer
> Case Western Reserve University
>
>
> On Thu, Mar 6, 2014 at 7:57 PM, David Wilson <davidwwilson at comcast.net
> >wrote:
>
> > Largest zeroless p-smooth number for the first few primes p.
> >
> >
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