log(m)'s close to integers, various log bases

[cloitre]

le 21/01/03 5:10, Neil Fernandez à primeness at borve.demon.co.uk a écrit :

> In message <Pine.LNX.4.44.0301202025080.17633-100000 at eva117.cs.ualberta.
> ca>, Jim Nastos <nastos at cs.ualberta.ca> writes
>> On Tue, 21 Jan 2003, Neil Fernandez wrote:
>> 
>>>> 7881196 -
>>>> 12752043 +
>>>> 20633239 +
>>> 
>>> I get:
>>> 
>>> log_phi(20633239)=34.9999999999999951187742...
>> 
>> I confirm. Very strange. My same code now gives 20633239 -  ... I'm not
>> sure what happened. But now I'm getting the following consecutive "+"s ,
>> and Maple confirms they are both slightly above an integer. Someone should
>> confirm that these are indeed consecutive terms, though.
>> 
>> 3010349 -
>> 4870847 +
>> 7881196 -
>> 12752043 +
>> 20633239 -
>> 33385282 +
>> 87403803 +
> 
> I get the above too but 54018521 (-) should be on the list between the
> last two terms mentioned.
> 
> log_phi(33385282)=35.000000000000001864462320812981771451985064099516
> log_phi(54018521)=35.999999999999999287838764192951002474040378463974
> log_phi(87403803)=37.000000000000000272021386608164000836163367085174
> 
> Neil

It appears that this sequence 2,3,4,7,11... is actually Lucas sequence for
n>1 (a(n)=a(n-)+a(n-2))

3,4,7,11,18,29,47,76,123,199,322,521,843,1364,2207,3571,5778,9349,15127,2447
6,39603,64079,103682,167761,271443,439204,710647,1149851,1860498,3010349,487
0847,7881196,12752043,20633239,33385282,54018521,87403803

If so sign are always alterning.

letting f(x)=round(frac(log(x)/log(phi)))

a=3;b=4;for(n=1,300,c=b+a;a=b;b=c;print1(f(c),","))

0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,
0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,
0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,
0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,
0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,
0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,
0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,
0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,

Benoit Cloitre