A124183: 1,000 terms & .gif

[Dean Hickerson]

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