OFFSET
1,3
COMMENTS
Set-theoretic definition: Let D={1,H} where 1 represents the unity number (not the digit), and H represents the hyperoperation H_w(y,z) (see A054871). Let S[n] be a finite sequence of symbols d in D that represents the number n, such that S[n] = {{p,d}|p is the position of the symbol d}. a(n) = min{((#S[n])-1)/3}. Let J be a set of sets such that J={{j,1}|j is a natural number}. a(n) = min{#(S[n]\J)}.
Recursive definition: a(n) = min{a(w) + a(y) + a(z) + 1 | H_w(y,z)=n}.
Conjecture: Let b(k) be a sequence such that b(k) = max{n|a(n)=k}. b(0)=1; b(1)=2; b(2)=3; b(3)=4; b(k)=H_(b(k-4))(3,2).
Let C be the smallest set of symbols such that c is an element of C if no other element or sequence of elements of C can be substituted for it.
Let R be a set of symbols r such that R={1,B}: where 1 represents the unity number (not the digit), and B represents a symbol of C. In particular Bm is the m-th symbol of C.
Let B1 be H, then D={1,B1}.
Let P(x) be a predicate about the natural number x.
Let T[P] be a finite sequence of symbols r that represents the property P, such that T[P] = {{q,r}|q is the position of the symbol r}.
Conjecture: Let k = min{h|h is a natural number, h >= (min{#T[P]}-1)/4} b(k) >= min{x|P(x) is true}).
LINKS
Alessandro M. Pettinari, Table of n, a(n) for n = 1..10000
Alessandro M. Pettinari, C program for n<65536
EXAMPLE
Here Hwyz = H_w(y,z):
1=1, a(1)=0;
2=H111, a(2)=1;
3=H1H1111, a(3)=2;
4=H1H111H111, a(4)=3;
5=H1H1H1111H111, a(5)=4;
6=HH111H1H1111H111, a(6)=5;
...
PROG
(C) // See Links.
CROSSREFS
KEYWORD
nonn
AUTHOR
Alessandro M. Pettinari, Nov 14 2025
STATUS
approved