# [seqfan] Gilbreath mysteries

Neil Sloane njasloane at gmail.com
Tue May 9 20:43:38 CEST 2023

(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="
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?