[seqfan] Re: A051264 == A050278 ?

Giovanni Resta g.resta at iit.cnr.it
Wed Jan 11 00:59:55 CET 2012

On 11/01/2012 0.53, Hans Havermann wrote:
> Giovanni Resta:
>> Since a number cannot be 9-persistent, I have set a(9)=0.
> Hmm. One of the first exercises in the Honsberger reference (quoted by
> Richard Guy previously in this thread) is "Prove that there exists at
> least one n-persistent number for
> each positive integer n."
> Are you sure you are calculating these correctly?

The problem is in the definition.

If one uses a strict definition of k-persistency (like you did in your 
sequence), i.e.    n is k-persistent if  1*n, 2*n, ..., k*n are 
pandigital, BUT (k+1)*n IS NOT,
then it is easy to see that there cannot be 9-persistent numbers.
In fact, if   n, 2*n, 3*n, ...., 9*n are all pandigital (and thus
n is a candidate to 9-persistency) then also 10*n, which is n plus a 0,
is also pandigital, hence our candidate n is at least 10-persistent.


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