[seqfan] Re: possible sequence

David Wilson davidwwilson at comcast.net
Sat Mar 8 22:20:47 CET 2014


Statistically, I think we should expect the number of zeroless p-smooth
numbers be finite for any p.

The finitude of A003001 would immediately follow from the finitude of
zeroless 7-smooth numbers.

> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Charles
> Greathouse
> Sent: Thursday, March 06, 2014 9:32 PM
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Re: possible sequence
> 
> A better one: 2^25 * 3^227 * 7^28 has 140 digits. Any improvements would
> have to be > 10^250.
> 
> Charles Greathouse
> Analyst/Programmer
> Case Western Reserve University
> 
> 
> 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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