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

Alexander Povolotsky apovolot at gmail.com
Thu Nov 6 23:52:17 CET 2008

```All terms of this sequence seems to have common factor of 3.
If so, than the sequence could be reduced to
26, 260, 299, 306, 393, 455, 592, 858, 866, 989, 1056, 1254, 1586, 1925, 1932

ARP
================================================
On Thu, Nov 6, 2008 at 3:27 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. "
> The proof is easy and I did it in four cases.
> So all numbers of the form 10^m*78, 10^m*897, 10^m*918, 10^m*1179, 10^m*1365,
> ... are in the sequence.
>
> Thanks,
> Farideh
>
>
> Quoting David Wilson <dwilson at gambitcomm.com>:
>
>> The n with
>>
>> n = a concat b, phi(n) = phi(a) * phi(b)
>>
>> seem to be fairly common, e.g:
>>
>> phi(78) = phi(7)*phi(8)
>> phi(780) = phi(7)*phi(80)
>> phi(897) = phi(89)*phi(7)
>> phi(918) = phi(91)*phi(8)
>> phi(1179) = phi(11)*phi(79)
>>
>> The list starts
>>
>> 78 780 897 918 1179 1365 1776 2574 2598 2967 3168 3762 4758 5775 5796
>> 7800 7875 7917 8217 8970 9180 9576 11790 13650 13662 13875 13896 14391
>> 17760 18564 18858 19812 20097 25740 25935 25974 25980 27573 28776 28779
>>
>> Maximilian Hasler wrote:
>>> (10:05) gp >
>>> for(i=11,9999,i%10|next;eulerphi(i)==eulerphi(i\10)*eulerphi(i%10)&print1(i","))
>>> 78,897,918,2598,4758,7917,8217,
>>>
>>> Maximilian
>>> PS: can one have eulerphi(i)=product(eulerphi( k-th digit of i )) for i>78 ?
>>> I don't think so; it seems as if there would be strict ">" for all i
>>> different from 78.
>>>
>>>
>>
>>
>> _______________________________________________
>>
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>>
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> University of Isfahan (http://www.ui.ac.ir)
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```