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

Maximilian Hasler maximilian.hasler at gmail.com
Fri Nov 7 00:47:58 CET 2008

```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)
>>
>>
>>
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>>
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>>
>
>
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```