A124183: 1,000 terms & .gif

[zak seidov]

Oh, My Gods!!

Due to my silly error in my code, both
file and graph were WRONG!

As Hugo complaints that I send attachments,
I don't know how to show correct ones.

So I'll send them only to Dean (in a separate
message).

Also I send to Neil 
(in a separate message)
correct b124183.txt
with apologies!

Dean, can you send %C A124183 to Neil?

Thank you very much for your
really "math" message.

WADR, Zak  

--- Dean Hickerson <dean at math.ucdavis.edu> wrote:

> zak seidov wrote:
> 
> > I send the b124183.txt, and additionally,
> > (for seqfans convenience, before it appeared in
> OEIS) A124183.gif.
> ...
> > 0   0
> > 1   1
> > 2   2
> > 3   611
> > 4   643
> > 5   713
> > 6   623
> ...
> 
> The terms from a(3) on are wrong.  The sequence
> starts with 0, 1, 2, 4, 11,
> 8, 6, 15, 13, 19, 19.  But the graph looks about
> right, so this must be
> a transcription error rather than a computation
> error.
> 
> > My Q to seqfan gurus:
> > What's an origin of two straight line patterns, in
> graph? 
> 
> The definition of the sequence is:
> 
>     a(0) = 0; a(1) = 1; for n >= 2, a(n) = the nth
> integer from among those
>     positive integers which are coprime to
> a(n-1)+a(n-2).
> 
> From a(7) on, all terms are odd, so a(n-1)+a(n-2) is
> even for n>=9.  Hence
> the n-th integer coprime to it is at least 2n-1. 
> And if a(n-1)+a(n-2)
> happens to be 2 times a prime, then a(n)=2n-1.  This
> happens often enough
> to explain the lower line.  More generally, if
> a(n-1)+a(n-2) is a small
> power of 2 times a prime, then a(n) will be just
> slightly larger than 2n-1.
> The resulting points in the graph are close enough
> to the lower line so they
> look like they're part of it.
> 
> If a(n-1)+a(n-2) is 6 times a large prime, then
> about 1/3 of all numbers
> are coprime to it, so a(n) will be about 3n.  That
> explains the next most
> visible 'line' in the graph.  It's actually two
> lines, a(n)=3n+1 and
> a(n)=3n+2.  And again there are points close to it,
> caused by a(n-1)+a(n-2)
> being a small power of 2 times a small power of 3
> times a prime.
> 
> No doubt the other, slightly less visible, lines
> have similar explanations.
> 
> Probably a(n)/n is unbounded:  If the terms are
> sufficiently random, then
> there should be values of n for which a(n-1)+a(n-2)
> is divisible by many
> small primes, making a(n)/n large.  (For example, 
> a(1125)+a(1124) =
> 4259+3931 = 2 * 3^2 * 5 * 7 * 13,  and  a(1126)/1126
> = 5339/1126 = 4.74...)
> But I won't try to guess an asymptotically accurate
> upper bound for a(n).
> 
> Dean Hickerson
> dean at math.ucdavis.edu
> 



 
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