# [seqfan] Re: A051264 == A050278 ?

Richard Guy rkg at cpsc.ucalgary.ca
Mon Jan 9 23:46:27 CET 2012

Ross Honsberger, More Math.. Morsels, MAA, 1991, p.15, reads

Morsel 6   Persistent Numbers   If a positive integer $k$
contains all ten digits, 0, 1, 2,\ldots, 9, and this property
persists through all its mutiples $k$, $2k$, $3k$, \ldots, then
$k$ is said to be a {\em persistent} number. It turns out that
there are no persistent numbers: the demand for persistence
through {\em all} multiples is just too much to ask.  An
unbroken initial run of successes, however, does not go
unrecognized; if the first $n$ multiples $k$, $2k$, \ldots, $nk$,
each contain all ten digits, then $k$ is awarded the distinction
of being called {\em n-persistent}.  For example, $k=1234567890$
is 2-persistent because $2k=2468135780, but not 3-persistent, in view of$3k=3703703670$. Some integers manage to hold out for a long time; the number$k=526315789473684210$is 18-persistent, but not 19-persistent. Prove that there exists at least one$n$-persistent number for each positive integer$n\$.

R.

On Mon, 9 Jan 2012, franktaw at netscape.net wrote:

> Can anyone access the book "More Mathematical Morsels" by R. Honsberger,
> referenced in the MathWorld article? It presumably has a definition of
> n-persistent, and as the published version, we should match it. (The
> reference in the article just took me to Amazon.com.) This may also bear on
> the claim David is questioning in the MathWorld article.
>
> Note that the claim that there are no infinity-persistent numbers is strongly
> validated by A079339.
>
>
> -----Original Message-----
> From: David Wilson <davidwwilson at comcast.net>
>
> I think your "exactly" interpretation is the correct one, otherwise
> A051264 would coincide numerically with either A050278 or A171102 in the
> OEIS, which it does not starting at its first term. I would not expect
> such a mistake from the author.
>
> Given the "exactly" interpretation, the MathWorld claim that there
> exists a k-persistent number exists for each k becomes interesting and
> begs both a proof and an OEIS sequence of the smallest k-persistent
> number for each k.
>
> On 1/9/2012 3:22 AM, franktaw at netscape.net wrote:
>> From the definitions, A051264 should be the same as A171102, not
> A050278.
>>
>> However, it appears that n-persistent is being used in a slightly
>> different way from what the MathWorld article defines. It seems that
>> it is being used to mean that the number is n-persistent as defined
>> but not (n+1)-persistent - one could perhaps call this exactly
>> n-persistent. Based on that, A051264 is different from the other two
>> sequences.
>>
>> Certainly the definitions need to be clarified. The other sequences
>> referenced in the MathWorld article also appear to be defined in this
>> manner.
>>
>>
>> -----Original Message-----
>> From: Moshe Levin <moshe.levin at mail.ru>
>>
>> Shouldn't  A051264 (1 - persistent numbers)
>> coincide with A050278 (Pandigital numbers)?
>>
>>    My friend wrote to the author (of both seqs), in vail,
>>
>> I think that A051264 == A050278 by definition.
>>
>> Thanks,
>> ML
>>
>>
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