[seqfan] Re: Gilbreath mysteries

[Thomas Scheuerle]

Dear discussion list,

I just learned by this mail from the existence of these fascinating facts.
I will ask some likely stupid questions, as I did not yet read
much about this phenomenon.
What first surprised me is how it is yet unproven.
Sometimes if you start thinking about naive approaches to prove something
you will learn why it is difficult and thus obtain deeper understanding
of the underlying problem itself.

The first thing which was obvious to me
is that all numbers in the first column of A036262 below 2 must be odd,
and all other columns below the primes can only be even.

We get only ones in the first column if in the second column
is no number > 2. 

Now my first question: 
By intuition it felt to me that this condition will be always met
if abs(prime(m+1)-prime(m)) <= 2*m.
Is this true ?

My second question:
abs(prime(m+1)-prime(m)) <= 2*m appears to me intuitively true,
because primes are generated by a sieve, and each new obtained prime in this sieving process can increase the gap to the next found prime maximally by two, so we would expect that abs(prime(m+1)-prime(m)) <= 2*m is true.
Is this true ?

Regarding question 2, let us replace the primes with something else:
2,3,7,13,21,31,43     2, A002061
1,2,4,6,8,10,12
1,2,2,2,2,2 ,2,
1,0,0,0,0,0,0
1,0,0,0,0,0,0
1,0,0,0,0,0

If we have a slower growing sequence of odd numbers in the first row,
will we still see all ones in the first column ? 
What kind of sequences could be the exceptions ?

best regards

Thomas

> (Apologies for double posting, but I would really like to know the answers)
> 
> 1. In 1958 the magician Norman L. Gilbreath observed that if you construct
> an array in which the first row are the primes and subsequent rows are the
> absolute values of the differences of the previous row,
> 
> you get an array (A036262)
> 
> 
> 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101
> ...
> 
> 1 2 2 4  2  4  2  4  6  2  6  4  2  4  6  6  2  6  4  2  6  4  6  8  4   2
> ...
> 
> 1 0 2 2  2  2  2  2  4  4  2  2  2  2  0  4  4  2  2  4  2  2  2  4  2   2
> ...
> 
> 1 2 0 0  0  0  0  2  0  2  0  0  0  2  4  0  2  0  2  2  0  0  2  2  0   0
> ...
> 
>  ...
> 
> 
> in which the first column appears to be 2 1 1 1 1 .... This is astonishing
> but has never been proved. See R. K. Guy, Unsolved Problems in Number
> Theory, Sec. A10. This was already noticed by Proth in 1878,
> 
> F. Proth, <a href="
> http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN598948236_0004&DMDID=DMDLOG_0076&IDDOC=630831">Sur
> la série des nombres premiers</a>, Nouv. Corresp. Math., 4 (1878)
> 236-240.who gave what is said to be a false proof.
> 
> 
> 2. If S is a sequence with offset o ("oh"), I define its Gilbreath
> transform GT(S) (with the same offset) in the analogous way (see A362451
> for precise definition). The Gilbreath conjecture is that GT(primes) =
> 2,1,1,1,...
> 
> 
> 3. Ten days ago Wayman Eduardo Luy and Robert G. Wilson observed in effect
> that GT(tau) (where tau is the number of divisors function) appears to be a
> sequence of 0's and 1's. This was also a surprise, and also seems to be
> unproved. See A361897, A362450, A462453, A362454.
> 
> 
> This led me to look at other number-theoretic sequences.
> 
> 
> 4. The sum of divisors function sigma(n) (A000203). The Gilbreath transform
> (A362451) begins
> 
> 
> 1, 2, 1, 1, 1, 2, 1, 0, 0, 1, 0, 4, 0, 3, 0, 2, 1, 1, 1, 0, 1, 1 ...
> 
> 
> and appears to be mostly 0's and 1's, interrupted by occasional geysers of
> increasing height and duration (see A362456, A362457). Is there any
> explanation for this?  The sum of aliquot parts function A001065 has a
> similar Gilbreath transform to sigma (see A362452).
> 
> 
> 5. It has been suggested that having G.T. equal to 1,1,1,... should hold
> for other sequences with similar properties to the primes. Does anyone know
> of other examples?
> 
> 
> 6. The Lucky numbers A000959 are similar to the primes in some ways. But
> the G.T. of the Lucky numbers is A054978, which begins
> 
> 
> 1, 2, 2, 0, 0, 0, 2, 2, 2, 0, 0, 0, 2, 0, 2, 0, 0, 2, 0, ...
> 
> 
> If the initial 1 is deleted and the terms are divided by 2, we get what
> appears to be a 0,1 sequence A362460. Why is this?
> 
> 
> 
> 7. The G.T. of the Euler phi function A000010 appears to be the period-2
> sequence 1,0,1,0,... (see A362913). Is this obvious? Or is it as mysterious
> as the Gilbreath conjecture?
> 
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-- 
T.S. <ts181 at mail.ru>