[seqfan] Re: D(n)

David Sycamore djsycamore at yahoo.co.uk
Sun Dec 8 23:07:16 CET 2019


Hi Ami, 
Thanks  for the corrections. Your numbers differ from mine from a(3) onwards, rather than a(4). If I am not mistaken a(n) and a*k(n),k =1,2... must each have only one pair of identical digits, and no number >2 of same digits, otherwise the number of steps to a stationary number will be reduced. Also I think that the size of the digits of a(n) must not descend within the string, otherwise it can’t be the “least k”. Therefore the digits must ascend in order (apart from the repeated ones); eg 4468, rather than 4486

Combining the above with your results, the updated terms are now (up to a(7)):

1, 11, 112, 166, 886, 4468, 22468, 112468...

I’m not yet certain that these really are the least possible numbers, and I’m also not sure about what you found for a(8,9,10). For example, your a(8)=116486, gives: 116486, 21248, 4148, 818, 161, 26 which is 5 steps. 

There are still only 8 terms up to now,  and I can’t find any more, so perhaps  this sequence may be finite after all; would you agree? 
Best
David. 

> On 8 Dec 2019, at 19:56, Ami Eldar <amiram.eldar at gmail.com> wrote:
> 
> Hello David,
> 
> I am getting different terms, starting from the 4th term:
> 1, 11, 112, 166, 886, 4486, 22486, 112486, 166486, 488686, 4486486,...
> 
> 
> 
> On Sun, Dec 8, 2019 at 7:16 PM David Sycamore via SeqFan <
> seqfan at list.seqfan.eu> wrote:
> 
>> Hello Seqfans,
>> For n =d_1 d_2...d_k, define D(n) to be the unique number formed by adding
>> all identical digits of n in order of appearance in n. (See draft A330273
>> for details). Example D(224)=44 and D(44)=8.
>> 
>> Let e(n) be the smallest number such that D*e(n)(n) can be reduced no
>> further (then e(224)=2). If the digits of n are all distinct, then e(n)=0,
>> since D(n)=n.
>> 
>> Let a(n) be the smallest number k such that e(k)=n; (n>=0). Does a(n)
>> always exist?
>> 
>> I am trying to find some terms. So far I have candidates for a(n) up to
>> n=7. (by hand) :
>> 
>> 1, 11, 112, 1124, 11248, 122468, 2486, 122468..... ?
>> 
>> But some of these may be wrong, I’m not sure....
>> 
>> If anyone feels like computing the right numbers please have a go
>> Thanks,
>> David.
>> 
>> 
>> 
>> 
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>> 
> 
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