Stern's diatomic series - frequencies

[franktaw at netscape.net]

It's not surprising that the most common values are prime.  Note
the property of A002487 that each positive rational occurs once
as a ratio of consecutive terms - or in other words, each pair of
relatively prime positive integers occurs once.  Since primes have
a higher density of relatively prime numbers, one would expect
to see them more often.

However, I would expect the most common value through the
first n values to gradually increase, hitting at least many of the
primes on the way.  The sudden jump to 71 is what surprises
me.

It does look like maybe primes one less than highly composite
numbers are favored - at least, that applies to both 71 and
419.  But why?  And why 71 more than, say, 59?

Franklin T. Adams-Watters


-----Original Message-----
From: eclark at math.usf.edu

  On Wed, 13 Sep 2006 franktaw at netscape.net wrote:

> This is A002487.
>
> I was looking at the frequencies of different numbers in
> this sequence.  Every odd prime from 3 through 29 takes
> a turn at being the term that occurs most often in an
> initial segment of the sequence (although not perfectly
> in order).  Then at n = 6122, suddenly 71 becomes the
> most common, and stays there through n = 16384
> (which is as far as I looked).
>
> What's so special about 71?
>
> What happens for larger values of n?  Is the most common
> value ever composite?
>

Well, I computed the most frequent values in a(0), a(1),a(2),...,a(10^6)

and found that among the top 73 most frequent values all were prime.
I think something strange is going on here.

Here's the distribution for these terms. The most frequent is 419 which
occurs 795 times.

[[419, 795], [701, 767], [619, 761], [433, 759], [661, 753], [431, 
753],
[479, 749], [839, 746], [647, 743], [653, 741], [379, 739], [461, 739],
[337, 738], [587, 737], [499, 736], [557, 735], [541, 734], [373, 734],
[523, 732], [463, 729], [547, 728], [421, 728], [577, 728], [389, 728],
[397, 727], [823, 726], [613, 725], [563, 724], [503, 723], [733, 723],
[359, 722], [449, 717], [607, 717], [467, 716], [241, 716], [599, 715],
[941, 715], [443, 715], [509, 711], [367, 711], [271, 711], [439, 709],
[487, 706], [643, 706], [677, 705], [787, 705], [821, 705], [673, 704],
[631, 704], [383, 703], [313, 703], [349, 702], [811, 701], [353, 700],
[307, 700], [457, 698], [521, 696], [659, 696], [773, 696], [401, 696],
[409, 696], [683, 695], [331, 695], [641, 695], [239, 695], [907, 695],
[797, 694], [347, 693], [691, 693], [571, 693], [263, 692], [251, 689],
[569, 688]]


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