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

David Wilson davidwwilson at comcast.net
Sun Sep 7 23:48:13 CEST 2014 

Using double precision arithmetic and C++ <cmath>, I get

0, 1, 260515, 37362253, 122925461, 534483448, ...

I am not sure how accurate C++ tan(x) is for very large x, so this would have to be verified in some arbitrary precision package.

> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Frank
> Adams-Watters
> Sent: Saturday, September 06, 2014 7:59 PM
> To: seqfan at list.seqfan.eu
> Subject: [seqfan] Re: Floor[Tan[n]]
> 
> I'll say. Up to 1 million I get:
> 
> 1, 260515
> 
> https://oeis.org/A088306 is related.
> 
> Franklin T. Adams-Watters
> 
> -----Original Message-----
> From: David Wilson <davidwwilson at comcast.net>
> 
> 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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