[seqfan] Re: Iterates of n - A002828(n) ?
Antti Karttunen
antti.karttunen at gmail.com
Wed Aug 21 11:18:54 CEST 2013
On Wed, Aug 21, 2013 at 11:59 AM, Antti Karttunen <antti.karttunen at gmail.com
> wrote:
>
> (Trying posting this again, hopefully without random blue regions, thanks
> to Gmail-formatting...)
>
> On Tue, Aug 20, 2013 at 7:51 PM, <seqfan-request at list.seqfan.eu> wrote:
> >
> > Send SeqFan mailing list submissions to
> > seqfan at list.seqfan.eu
>
>
> >
> > ------------------------------
> >
> > Message: 9
> > Date: Tue, 20 Aug 2013 12:51:00 -0400
> > From: Charles Greathouse <charles.greathouse at case.edu>
> > To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> > Subject: [seqfan] Re: Iterates of n - A002828(n) ?
> > Message-ID:
> > <
> CAAkfSG+9_jqc2US6533SAndiVzmJX_xonKVVS6kzNE6bvgT_fg at mail.gmail.com>
> > Content-Type: text/plain; charset=ISO-8859-1
> >
> > The conjecture is true. It suffices to prove it for m^2, m^2+1, and m^2+2
> > since all larger numbers will reduce to one of these. m^2 is the sum of
> one
> > square, so that takes it to m^2-1 as desired. m^2+1 is the sum of two
> > squares m^2 and 1^2, so that goes to m^2-1 for m > 0. m^2+2 is the sum of
> > three squares, m^2 + 1^2 + 1^2, so it will go to at least m^2-1.
>
>
> Thanks Charles. It is easy for those who remember that also 1 is a square!
>
> Obviously this generalizes also to http://oeis.org/A002376 (Least number
> of positive cubes needed to represent n) and higher powers as well.
>
> And now the thematic connection with A179016 and its kin is even clearer.
> It's just that A000120 = The least number of powers of 2 that sum to n.
> A007895 = The least number of Fibonacci numbers that sum to n.
> A034968 = The least number of factorials that add to n.
>
> So, I guess we could construct such sequences almost from any sequence
> Axxxxxx which is about monotone and contains one. E.g. A000290 and A000578
> for the squares and cubes cases, A000079, A000045 and A000142 for binary,
> Zeckendorf and factorial expansion beanstalks.
>
> So, I wonder, what would come from http://oeis.org/A014420
> (Minimum number of Catalan numbers to represent n.)
> (BTW, I think it would be just a good etiquette when one creates
> parameterized sequences like
> http://oeis.org/A161227 - http://oeis.org/A161236 that there would be a
> crossref-link to the most basic case of the set. Thanks!)
> or http://oeis.org/A057945 (Number of triangular numbers needed to
> represent n with greedy algorithm.) Is the "with greedy algorithm"
> qualification really necessary in the name?
>
> And from A008578 also http://oeis.org/A072491 which I guess is about the
> same as "The least (?) number of noncomposites needed to represent n".
>
> I guess we can leave 1 off in this case, and consider
> https://oeis.org/A051034
> (Minimal number of primes needed to sum to n.)
>
> I can't find "1,1,1,2,1,2,1,2,2,2,1,2,2,2,1,..." which is my quick
> hand-calculation for minimal number of partition numbers (
> https://oeis.org/A000041 ) that add to n. (Yes, this is already somewhat
> artificial.)
>
My intuition says that sequences concerning the least number of (2^k)-1's
(A000225) that sum to n is more fruitful. Probably one of these sequences
(all seem to be interesting ones):
https://oeis.org/search?q=1%2C2%2C1%2C2%2C3%2C2%2C1%2C2%2C3%2C2%2C3&sort=&language=&go=Search
Cheers,
Antti
>
>
> Terveisin,
>
> Antti
>
>
>
> >
> > Charles Greathouse
> > Analyst/Programmer
> > Case Western Reserve University
> >
> >
> > On Tue, Aug 20, 2013 at 10:26 AM, Antti Karttunen <
> antti.karttunen at gmail.com
> > > wrote:
> >
> > > On Mon, Aug 19, 2013, *L. Edson Jeffery* <lejeffery2 at gmail.com>
> > > <seqfan%
> > >
> 40list.seqfan.eu?Subject=Re%3A%20%5Bseqfan%5D%20Re%3A%20Fwd%3A%20Iterates%20of%20n%20-%20A002828%28n%29%20%3F&In-Reply-To=%3CCAGRLqMgJ1BunNuV3BuK1Os91Dm_L-%3Djy_7PnNuNQOAm_Hx1yiw%40mail.gmail.com%3E
> > > >
> > > wrote:
> > >
> > > >
> > > >
> > > > *in *http://list.seqfan.eu/pipermail/seqfan/2013-August/011586.html
> > > >
> > >
> > >
> > > >
> > > > - Previous message: [seqfan] Fwd: Iterates of n - A002828(n) ?<
> > > http://list.seqfan.eu/pipermail/seqfan/2013-August/011582.html>
> > > > - *Messages sorted by:* [ date ]<
> > > http://list.seqfan.eu/pipermail/seqfan/2013-August/date.html#11586>
> > > > [ thread ]<
> > > http://list.seqfan.eu/pipermail/seqfan/2013-August/thread.html#11586>
> > > > [ subject ]<
> > > http://list.seqfan.eu/pipermail/seqfan/2013-August/subject.html#11586>
> > > > [ author ]<
> > > http://list.seqfan.eu/pipermail/seqfan/2013-August/author.html#11586>
> > > >
> > > > ------------------------------
> >
> > > >
> > > > AK>...will the trajectory visit every square-1 (A005563) < k before
> it
> > > > reaches zero?
> > > >
> > > > Is it sufficient to prove, for any n, that max(square-1 < n) is in
> the
> > > > trajectory?
> > > >
> > > > Yes, just that.
> > >
> > > >
> > > > AK>Which would imply that there would be only one infinite sequence
> a_n
> > > > such that a(n-1) = a(n) - the least number of squares that add up to
> > > a(n).
> > > >
> > > > This should be essentially the first column in the first triangle
> below.
> > > >
> > > > AK>Then, would there be any regularity in the derived sequence
> counting
> > > the
> > > > number of iterations needed to hop from A005563(n+1) to A005563(n)?
> > > >
> > > > According to the second triangle below, it appears that your hopping
> > > > sequence should be {1,1,2,2,4,4,5,...}, unless I misunderstood or
> > > > miscalculated.
> > > >
> > > > Yes, exactly that, the first differences of the row lengths 1, 1, 3,
> 5,
> > > 9,
> > > 13,18, ...
> > > of the table given below.
> > >
> > > Not sure exactly what you mean by "regularity."
> > > >
> > > >
> > > Something like what is present in these https://oeis.org/A213709 &
> > > https://oeis.org/A226060 & https://oeis.org/A218543
> > > Some regularity (e.g. partial scale-invariance) mixed with some
> "random" or
> > > not-so-easily computable aspects.
> > >
> > >
> > > > Nice ideas, Antti.
> > > >
> > > >
> > > Not wholly my idea. I just tried to find a variant with a more
> > > "number-theoretical taste to it" than Carl White's original "binary
> > > beanstalk" sequence https://oeis.org/A179016 (to which the above three
> > > seq-links are related to) and other examples of such sequences that I
> have
> > > submitted in other bases.
> > > However, while it is easy to guarantee the existence of "choking
> points"
> > > (through which all the iterated paths will come through) in
> base-related
> > > beanstalk sequences, it seems not that easy when trying to construct
> such
> > > phenomena from sequences related to the traditional number theory.
> > > But what I glean from Pari-code at https://oeis.org/A002828
> > > it seems actually to be a some kind of "crypto-base" sequence.
> > > (Or at least involving modulus some power of 2. I have to re-read my
> > > Burton.)
> > >
> > >
> > > -- Antti
> > >
> > > PS: If anybody comments on this, please add CC: to my gmail-account,
> so I
> > > will get the reply even before the mailing list digest is ready.
> > >
> > > >
> > > > n Seq of trajectories for 0 <= n <= 48
> > > > --- -----------------------------------------
> > > > 0 0
> > > > 1 0
> > > > 2 0
> > > > 3 0
> > > > 4 3,0
> > > > 5 3,0
> > > > 6 3,0
> > > > 7 3,0
> > > > 8 6,3,0
> > > > 9 8,6,3,0
> > > > 10 8,6,3,0
> > > > 11 8,6,3,0
> > > > 12 9,8,6,3,0
> > > > 13 11,8,6,3,0
> > > > 14 11,8,6,3,0
> > > > 15 11,8,6,3,0
> > > > 16 15,11,8,6,3,0
> > > > 17 15,11,8,6,3,0
> > > > 18 16,15,11,8,6,3,0
> > > > 19 16,15,11,8,6,3,0
> > > > 20 18,16,15,11,8,6,3,0
> > > > 21 18,16,15,11,8,6,3,0
> > > > 22 19,16,15,11,8,6,3,0
> > > > 23 19,16,15,11,8,6,3,0
> > > > 24 21,18,16,15,11,8,6,3,0
> > > > 25 24,21,18,16,15,11,8,6,3,0
> > > > 26 24,21,18,16,15,11,8,6,3,0
> > > > 27 24,21,18,16,15,11,8,6,3,0
> > > > 28 24,21,18,16,15,11,8,6,3,0
> > > > 29 27,24,21,18,16,15,11,8,6,3,0
> > > > 30 27,24,21,18,16,15,11,8,6,3,0
> > > > 31 27,24,21,18,16,15,11,8,6,3,0
> > > > 32 30,27,24,21,18,16,15,11,8,6,3,0
> > > > 33 30,27,24,21,18,16,15,11,8,6,3,0
> > > > 34 32,30,27,24,21,18,16,15,11,8,6,3,0
> > > > 35 32,30,27,24,21,18,16,15,11,8,6,3,0
> > > > 36 35,32,30,27,24,21,18,16,15,11,8,6,3,0
> > > > 37 35,32,30,27,24,21,18,16,15,11,8,6,3,0
> > > > 38 35,32,30,27,24,21,18,16,15,11,8,6,3,0
> > > > 39 35,32,30,27,24,21,18,16,15,11,8,6,3,0
> > > > 40 38,35,32,30,27,24,21,18,16,15,11,8,6,3,0
> > > > 41 38,35,32,30,27,24,21,18,16,15,11,8,6,3,0
> > > > 42 39,35,32,30,27,24,21,18,16,15,11,8,6,3,0
> > > > 43 40,38,35,32,30,27,24,21,18,16,15,11,8,6,3,0
> > > > 44 41,38,35,32,30,27,24,21,18,16,15,11,8,6,3,0
> > > > 45 43,40,38,35,32,30,27,24,21,18,16,15,11,8,6,3,0
> > > > 46 43,40,38,35,32,30,27,24,21,18,16,15,11,8,6,3,0
> > > > 47 43,40,38,35,32,30,27,24,21,18,16,15,11,8,6,3,0
> > > > 48 45,43,40,38,35,32,30,27,24,21,18,16,15,11,8,6,3,0
> > > >
> > > > n n^2-1 Trajectories for n^2 - 1, 1 <= n <= 7
> > > > --- ------- -----------------------------------------
> > > > 1 0 0
> > > > 2 3 0
> > > > 3 8 6,3,0
> > > > 4 15 11,8,6,3,0
> > > > 5 24 21,18,16,15,11,8,6,3,0
> > > > 6 35 32,30,27,24,21,18,16,15,11,8,6,3,0
> > > > 7 48 45,43,40,38,35,32,30,27,24,21,18,16,15,11,8,6,3,0
> > > >
> > > >
> > >
> > > > Ed Jeffery
> > > >
> > > > -------------------------------------------------------------
> > > >
> > > AK wrote earlier:
> > >
> > > > Just wondering:
> > > >
> > > > If we iterate the function a(n) = n - A002828(n) (
> > > > https://oeis.org/A002828 ) from any starting value k, will the
> > > trajectory
> > > > visit every square-1 (A005563) < k before it reaches zero?
> > > > Which would imply that there would be only one infinite sequence a_n
> such
> > > > that a(n-1) = a(n) - the least number of squares that add up to a(n).
> > > >
> > > > Then, would there be any regularity in the derived sequence counting
> the
> > > > number of iterations needed to hop from A005563(n+1) to A005563(n) ?
> > > >
> > > >
> > > >
> > > > Cheers,
> > > >
> > > > Antti
> > > >
> > > >
> > > >
> > >
> > > _______________________________________________
> > >
> > > Seqfan Mailing list - http://list.seqfan.eu/
> > >
> >
> >
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