# [seqfan] Re: Floor[Tan[n]]

David Wilson davidwwilson at comcast.net
Sat Sep 6 22:59:58 CEST 2014

```Related sequence:

n for which [tan(n)] < n.

> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Alex
> Meiburg
> Sent: Thursday, September 04, 2014 10:33 PM
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Re: Floor[Tan[n]]
>
> It should be mentioned at least once in this discussion that the
> observation Pohjola made, that iterating the function always leads to 0 or
> 1, is far from certain, and would be fascinating if true. Although I
> verified it for numbers from -10^7 to 10^7, to prove it always true would
> require deep statements about the irrationality of pi/2 -- and the
> non-monotonicity of its behavior is reminiscent of the headache brought on
> by the Collatz conjecture. The statement that it always reaches one of
> these two, I would wager, would probably have difficulty on par with such
> conjectures, or interesting open problems like the convergence of
> http://mathworld.wolfram.com/FlintHillsSeries.html .
>
> From a "probabilistic" viewpoint it makes sense; a number n has O(1/n^2)
> probability of being the output of Floor[Tan[x]] for random chosen x, so
> the chance at any number, say, over 1000 produces itself as the output, is
> the sum 1/ 1000^2 + 1/1001^2 + 1/1002^2 ... which converges to a finite
> probability. This gives the notion that "statistically" there's a good
> chance of no fixed points. (With nothing to say about possible 2-loops or
> 3-loops.) Indeed, if you looked at the behavior of Floor[Tan[pi*a*n]] for
> some irrational a, choosing a randomly would lead to this probability of
> encountering fixed points. This doesn't tell us much about what happens in
> the case of a = 1/pi, of course. :P
>
>
> -- Alexander Meiburg
>
>
> 2014-09-04 16:40 GMT-07:00 Frank Adams-Watters
> <franktaw at netscape.net>:
>
> > Every integer means every integer. If it meant every value that occurs in
> > the sequence, it would say so.
> >
> > But I don't see what your problem is. (2) implies (1); so if there is a
> > proof using (1), then (2) will be sufficient for that proof, too.
> >
> > Franklin T. Adams-Watters
> >
> >
> > -----Original Message-----
> > From: Veikko Pohjola <veikko at nordem.fi>
> > To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> > Sent: Thu, Sep 4, 2014 2:37 pm
> > Subject: [seqfan] Re: Floor[Tan[n]]
> >
> >
> > I come back to my original question. I still feel that I did not get the
> > To me (and actually to some of my colleagues) the statement "Evey [should
> > be
> > every, of course, but that is not the point] integer appears infinitely
> > often"
> > is most vague. What is referred to by "every integer"? (1) All integers
> > which
> > once appear in the sequence, or (2) all integers which exist? If it is the
> > (1),
> > then it is easy to understand (at least to me) why the sequence reduces to
> > the
> > limiting sequence composed of 0's and 1's when nesting the function
> > Floor[Tan],
> > otherwise not.
> > Veikko
> >
> > Veikko Pohjola kirjoitti 3.9.2014 kello 12.03:
> >
> >  The piece of sequence I gave is of  course that of the differences
> >>
> > between the
> > positions of 1's.
> >
> >> The positions themselves go like this
> >> 2, 5, 24, 27, 46, 49, 68, 71, 90, 93, 106, 112, 115, 122, 128, 134,
> >>
> > 137, …
> >
> >> (Not in OEIS)
> >> Veikko
> >>
> >>
> >>
> >> Veikko Pohjola kirjoitti 3.9.2014 kello 11.50:
> >>
> >>  I decided to pose these questions having played with another
> >>>
> >> sequence, which
> > forms when nesting Floor[Tan] to A000503 sufficiently many times. The end
> > result
> > is a sequence composed of 0’s and 1’s. The position on 1’s in this
> > sequence does
> > not seem to follow any regular pattern.
> >
> >>
> >>> After applying the Floor[Tan] 8 times to Floor[Tan[n]], n=0...10^5,
> >>>
> >> the
> > position of 1’s in the limiting sequence (not changing when applying the
> > function the 9th time) is as follows:
> >
> >> 3, 19, 3, 19, 3, 19, 3, 19, 3, 13, 6, 3, 7, 6, 6, 3, 7, 6, 6, 3, 13,
> >>>
> >> 6, 3, 3,
> > 10, 6, 3, 3, 10, 6, 3, 3, 16, 3, 3, 16, 3, 3, 19, 3, 7, 12, 3, 19, 3, 7,
> > 12, 3,
> > 19, ...
> >
> >> Interesting, huh?
> >>> Veikko
> >>>
> >>>
> >>> Neil Sloane kirjoitti 3.9.2014 kello 10.18:
> >>>
> >>>  well, tan(n) = tan(n +2Pi), right?
> >>>>
> >>>> and as n varies, n mod 2Pi will be dense in 0 to Pi
> >>>>
> >>>> On Wed, Sep 3, 2014 at 3:00 AM, Veikko Pohjola <veikko at nordem.fi>
> >>>>
> >>> wrote:
> >
> >> Dear seqfans,
> >>>>>
> >>>>> It is obvious that Tan[x] covers all real numbers (I guess). It
> >>>>>
> >>>> may be
> > obvious that Floor[Tan[x]] covers all natural numbers. But it is pretty
> > far from
> > obvious, to me, that even Floor[Tan[n]], where n is an integer from zero to
> > infinity, would also cover all natural numbers.
> >
> >>
> >>>>> Is it this last statement above, what is meant by the comment
> >>>>>
> >>>> ”Evey integer
> > appears infinitely often.”, which appears in A000503? If it is, could and
> > shouldn’t it be provided with a justification in the case of being a
> > conjecture,
> > or with a proof in he case of being a postulate?
> >
> >>
> >>>>> Veikko
> >>>>>
> >>>>> _______________________________________________
> >>>>>
> >>>>> Seqfan Mailing list - http://list.seqfan.eu/
> >>>>>
> >>>>
> >>>>
> >>>>
> >>>> --
> >>>> Dear Friends, I have now retired from AT&T. New coordinates:
> >>>>
> >>>> Neil J. A. Sloane, President, OEIS Foundation
> >>>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> >>>> Also Visiting Scientist, Math. Dept., Rutgers University,
> >>>>
> >>> Piscataway, NJ.
> >
> >> Phone: 732 828 6098; home page: http://NeilSloane.com
> >>>> Email: njasloane at gmail.com
> >>>>
> >>>> _______________________________________________
> >>>>
> >>>> Seqfan Mailing list - http://list.seqfan.eu/
> >>>>
> >>>
> >>>
> >>> _______________________________________________
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> >>>
> >>
> >>
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> >>
> >
> >
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