[seqfan] Re: Gilbreath mysteries

[Peter Munn]

On Wed, May 10, 2023 1:14 pm, Elijah Beregovsky wrote:
> Apparently, the conjecture that if the sequence starts with a 2, continues
> with odd numbers and grows slow enough, then its Gilbreath transform will
> be all 1 is false.
> https://11011110.github.io/blog/2011/02/20/anti-gilbreath-sequences.html
>

The final paragraph in that blog post has several interesting statements.
One, in particular, might be seen as setting a challenge OEIS could help
with: 'The same thing seems likely to be true for most "natural" sequences
[that meet the restrictions], sequences that aren't artificially defined
in such a way as to violate Gilbreath's conjecture.'

So I suggest we might try some counting. We could look at sequences "like
the primes", but I think it simpler to look at sequences that are "like"
the 1st differences, divided by 2, of the odd primes.

As a first idea how about this?: "Number of qualifying sequences, b, of
length n that have numbers greater than 1 in their Gilbreath transform,
where a sequence b qualifies if 1 <= b(i) <= 1 + floor(log(i)) for 1 <= i
<= n."

Best regards,

Peter