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

David Wilson davidwwilson at comcast.net
Fri Oct 31 13:10:19 CET 2008

```----- Original Message -----
From: "Maximilian Hasler" <maximilian.hasler at gmail.com>
To: "Sequence Fanatics Discussion list" <seqfan at seqfan.eu>
Cc: "Farideh Firoozbakht" <mymontain at yahoo.com>
Sent: Monday, October 27, 2008 10:34 AM
Subject: Re: [seqfan] Fwd: A nice (decimal) property of 78

> (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.

Let

phi(n) <= PROD(digit k of n; phi(k))

But for a decimal digit k, we must have phi(k) <= 6, so that

phi(n) <= 6^d.

which is to say

n PROD(distinct primes p dividing n; (p-1)/p) <= 6^d

But n has d digits, so n >= 10^(d-1), giving

10^(d-1) PROD(distinct primes p dividing n | n; (p-1)/p) <= 6^d,

or

PROD(distinct primes p dividing n; (p-1)/p) <= 10*(6/10)^d

I don't have time to do the calculations now, but you will find that for
sufficient d, you will need many primes on the left side to make the
inequality work, many more than can possibly divide the d-digit number n.
Given a suitable upper bound on p(n), I imagine a small maximum d can be
established rigorously. This would provide an upper limit on n.

> On Mon, Oct 27, 2008 at 05:01, Olivier Gerard <olivier.gerard at gmail.com>
> wrote:
>> Message from Farideh Firoozbakht <mymontain at yahoo.com>
>>
>> Please put Farideh in copy of your answers today.
>>
>> Olivier
>>
>>
>> ---------- Forwarded message ----------
>>
>> Dear seqfans,
>>
>> phi(78)=phi(7)*phi(8), does there exist another such number?
>>
>> Farideh
>>
>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/

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