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

David Wilson davidwwilson at comcast.net
Fri Nov 7 04:07:32 CET 2008

```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.

At any rate, I use the expressions

x == a (mod b)

to mean x is congruent to a modulo b, and

a mod b

to mean the unique 0 <= x < b with x == a (mod b).

Also, I've seen || used for string concatenation before, was it in PL/1?  It
would be convenient to have an ASCII operator for base 10 (or stated base)
numeric concatenation.

What would REALLY be convenient is for the OEIS to have an mathematical
symbol glossary.

----- Original Message -----
From: "Maximilian Hasler" <maximilian.hasler at gmail.com>
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Sent: Thursday, November 06, 2008 6:47 PM
Subject: [seqfan] Re: Fwd: A nice (decimal) property of 78

> up to 10^6, this is true for all numbers where the second part "b" has
> 1 or 2 digits, but not for 3 digits.
> Specifically,
> 43904,101794,565964,779779,811928,905905,925925,...
>
> are not multiples of 3 but verify phi(n) = phi( [ n/1000 ] ) * phi( n
> "mod" 1000)
>
> Maximilian
> PS: below you can find all numbers below 10^6 with the  phi( a||b ) =
> phi(a) phi(b) property - (unless there's some bug in my code...)
>
>
> for(i=11,10^6,i%10|next;eulerphi(i)==eulerphi(i\10)*eulerphi(i%10)&print1(i","))
>
> [78,897,918,2598,4758,7917,8217,18858,20097,25935,54678,61677,93738,152337,448218,670197,812175,994917]%3
> %122 = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
>
> for(i=101,999999,i%100|next;eulerphi(i)==eulerphi(i\100)*eulerphi(i%100)&print1(i","))
>
> [780,1179,1365,1776,2574,2967,3168,3762,8970,9180,9576,14391,18564,25974,25980,27573,28776,28779,33165,43362,44574,47580,47592,48573,49764,51576,54168,56574,69573,74598,78174,79170,79365,82170,102179,115776,132165,150597,163176,174168,177576,181374,188580,189774,200970,218367,229164,233367,259350,266574,272379,300768,416361,508368,514368,522165,528165,546780,616770,625374,647595,662598,692973,711171,747594,788985,810774,825588,937380]%3
> %123 = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0
> , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
> 0, 0, 0, 0]
>
>
> for(i=1001,999999,i%1000|next;eulerphi(i)==eulerphi(i\1000)*eulerphi(i%1000)&print1(i","))
>
> [5775,5796,7800,7875,11790,13650,13662,13875,13896,17760,19812,25740,29670,31680,35919,37620,43731,43904,49833,53742,65751,65793,67722,73584,73683,89700,91800,91845,95760,97662,101794,119841,121806,139608,143910,155727,161802,161805,179592,185640,185796,223812,229833,257847,259740,259800,275730,287760,287790,301758,305829,323697,325872,331650,335775,335856,353625,421689,433620,445740,451875,469752,475794,475800,475920,485730,493584,497640,511803,515760,517875,541680,541728,565740,565964,599847,629724,655875,673608,679776,695730,745923,745980,763812,779779,781740,791700,793650,793854,811928,817614,821700,889713,905905,925925,929604,973812]%3
> %126 = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0,
> 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
> , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
> 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0
> , 0, 0, 0, 2, 0, 0, 0, 1, 2, 0, 0]
>
> /* these are the nonzero ones among the above :*/
> for(i=1001,999999,i%1000|next;eulerphi(i)==eulerphi(i\1000)*eulerphi(i%1000)&
> i%3 & print1(i","))
> 43904,101794,565964,779779,811928,905905,925925,
>
>
> for(i=10001,999999,i%10000|next;eulerphi(i)==eulerphi(i\10000)*eulerphi(i%10000)
> & print1(i","))
> 57750,57960,59178,78000,78606,78750,79776,117900,118866,136500,136620,137085,137418,137646,137772,138381,138750,138960,139065,139212,176928,177600,196416,196875,197025,197136,197325,197334,198120,198288,198414,199044,257400,296700,316485,316800,318015,358875,359190,359751,376200,377952,378833,417252,417423,417571,417693,437310,439040,497925,498330,537420,617076,619685,657510,657657,657930,658983,659256,677220,717384,735042,735840,736830,738374,778932,778968,798036,896586,897000,899875,918000,918450,957600,958125,976620,
>
> for(i=100001,999999,i%100000|next;eulerphi(i)==eulerphi(i\100000)*eulerphi(i%100000)
> & print1(i","))
>
> 195195,577500,579600,588672,589155,591780,596751,597588,598212,771936,780000,780735,781326,786060,
> 787500,788196,793176,797760,799176
>
> ---
>
> On Thu, Nov 6, 2008 at 6:52 PM, Alexander Povolotsky <apovolot at gmail.com>
> wrote:
>> 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.
>>>>>
>>>>>
>>>>
>>>>
>>>> _______________________________________________
>>>>
>>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>> ----------------------------------------------------------------
>>> University of Isfahan (http://www.ui.ac.ir)
>>>
>>>
>>>
>>> _______________________________________________
>>>
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>
>>
>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/

--------------------------------------------------------------------------------

No virus found in this incoming message.
Checked by AVG - http://www.avg.com
Version: 8.0.175 / Virus Database: 270.9.0/1771 - Release Date: 11/6/2008
7:58 AM

```