[seqfan] Re: Fwd: A nice (decimal) property of 78

[f.firoozbakht at sci.ui.ac.ir]

Quoting Maximilian Hasler <maximilian.hasler at gmail.com>:

> On Thu, Nov 6, 2008 at 4:27 PM,  <f.firoozbakht at sci.ui.ac.ir> wrote:
>>> n = a concat b, phi(n) = phi(a) * phi(b)
>> I think your sequence (78, 780, 897, 918, 1179, 1365, 1776, 2574, 2598,
>> 2967, 3168, 3762, 4758, 5775, 5796, ...) is a nice sequence and it should
>> be submitted. Can you please do it?
>> I proved an interesting property of this sequence: " If n is in the
>> sequence then 10^m*n for each natural number m is also in the sequence. "
>> So all numbers of the form 10^m*78, 10^m*897, 10^m*918, 10^m*1179,
>> ... are in the sequence.
>
> If such a thing is the case, then usually it is a good idea to submit
> the sequence of "primitive elements" only (maybe as second, separate
> sequence).
> In fact it is somehow "much more interesting" in the sense that it
> easily allows to produce the "complete" sequence, while the latter
> will soon be filled up with terms derived from earlier terms (with
> many trailing zeroes) while "interesting" new elements become less and
> less frequent.
>
> (Btw. it is even not completely obvious to me, at a first glance,
> whether the sequence of such primitive elements will necessarily be
> infinite. Did you consider this question?)
>
> another generalization (in fact, in some sense also a bit "back to    
>    the roots"):
> allow for concatenation of more than just 2 numbers.
>
> Maximilian

Quoting David Wilson <davidwwilson at comcast.net>:

> I think maybe Hasler's sequence
> phi(n) = phi([n/10])phi(n mod 10)
> and my sequence
> phi(n) = phi([n/10^k])ph(n mod 10^k) for some k >= 1
> are probably interesting enough to merit OEIS inclusion. Whether moduli
> of 100, 1000, etc. are OEISworthy is debatable.
...


> (Btw. it is even not completely obvious to me, at a first glance,
> whether the sequence of such primitive elements will necessarily be
> infinite. Did you consider this question?)

Yes, I guess that the set of such primitive elements (members that
10 doesn't divide them) is infinite. We can try to prove it!

As you pointed out we can define and submit a generalized sequence of
concatenation of two or more numbers.
This sequence has the same property (If n is a term then 10*n is also in
the sequence). And the primitive elements is an interesting subsequence
of this sequence.

31977, 437388, 773976 & 778998 are four terms of the sequence that are
concatenation of more than two numbers.

phi(31977)=phi(31)*phi(97)*phi(7)
phi(437388)=phi(43)*phi(73)*phi(88)
phi(773976)=phi(7)*phi(73)*phi(976)
phi(778998)=phi(77)*phi(89)*phi(98)

Some primitive terms greater than 10^9:

phi(1000396644)=phi(100039)*phi(6644)
phi(1000857375)=phi(100085)*phi(7375)
phi(1001637585)=phi(100163)*phi(7585)

phi(10000279191)=phi(10000270*phi(9191)

***

Farideh







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