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

Sun Sep 7 01:44:37 CEST 2014

Well, actually the more interesting sequence is

Positive integers n for which tan(n) >= n

These are rare.

> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Frank
> Adams-Watters
> Sent: Saturday, September 06, 2014 5:46 PM
> To: seqfan at list.seqfan.eu
> Subject: [seqfan] Re: Floor[Tan[n]]
> 
> Except for 0, this is the same as:
> 
> n for which tan(n) <= n;
> 
> which seems more natural to me.
> 
> Franklin T. Adams-Watters
> 
> -----Original Message-----
> From: David Wilson <davidwwilson at comcast.net>
> To: 'Sequence Fanatics Discussion list' <seqfan at list.seqfan.eu>
> Sent: Sat, Sep 6, 2014 3:59 pm
> Subject: [seqfan] Re: Floor[Tan[n]]
> 
> 
> 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
> > > answer.
> > > 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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