# [seqfan] Re: Gilbreath mysteries

Elijah Beregovsky elijah.beregovsky at gmail.com
Wed May 10 14:14:18 CEST 2023

```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

I’ve gone through several well-known sequences and applied Gilbreath transform to them, here’s what I got (most of the following is conjectural, based on the initial terms):

A000108 (Catalan numbers) -> A005043 (Riordan numbers)
A000069 (odious numbers) -> A255738
A001969 (evil numbers) -> A255738 with a(1)=0, a(2)=3
A00142 (factorials) -> A000255 -> A087981
A000041 (partitions) -> A196251 by definition
A002110 (primorials) -> A121572 (subprimorials)
A001045 (Jacobsthal sequence) -> 0,1,1,1,1,1…
A000312 (n^n) -> abs(A069856)
A003309 (Ludic numbers) -> 1,1,0,1,0,1,0,1,0…
A007952 (sieve of Tchoukaillon) -> 0,1,1,1,1…

I wonder how many of these are obvious?
A thing that comes to my mind is that patterns of the Gilbreath transform may be somehow possible to elicit from the sieve that generated the initial sequence (if it was generated by a sieve)

There are also a bunch of famous sequences whose Gilbreath transforms are not in the OEIS (e.g. Kolakoski sequence, semiprimes, perfect numbers, taxicab numbers, Yellowstone permutation).

Cheers,
Elijah
```