Commonalities of Kolmogorov Topologic Spaces and Addition Chains

• forall n ∈ N:
  ℓ(n+1) ≤ ℓ(n)+1
  m(n+1) ≤ m(n)+1
• forall n ∈ N \ {1}:
  ℓ(n+2) ≤ ℓ(n)+1
  m(n+2) ≤ m(n)+1
• forall n,h ∈ N:
  ℓ(n*h) ≤ ℓ(n)+ℓ(h)
  m(n*h) ≤ m(n)+m(h)
• forall n ∈ N with ν(n) ∈ {1,2} :
  there is a Kolmogorov topologic space with n open sets and at least k points
  there is a shortest addition chain for n of length k.
• forall n ∈ N:
  ℓ(n) ≤ λ(n)+ν(n)-1
  m(n) ≤ λ(n)+ν(n)-1

Define t(k) = | {n ∈ N : m(n)=k } |, i.e. to be the number of sizes (their number of open sets) of finite topologies needing at least k points and define f(k) = min {n ∈ N : m(n)=k } , i.e. to be the minimal size (their number of open sets) of a finite topology needing at least k points. Remarkably it holds five times f(k)=2*f(k-1)-3, namely k ∈ { 4, 5, 6, 12, 14} and five times is f(k)+1 a power of two. namely k ∈ { 0, 2, 4, 10, 15}.
It follows a table to compare these sequences with the corresponding sequences c(k) and d(k) for shortest addition chains:

k	c(k)	s(c(k))	f(k)	d(k)	t(k)
0	1	0	1	1	1
1	2	0	2	1	1
2	3	1	3	2	2
3	5	1	5	3	3
4	7	2	7	5	5
5	11	2	11	9	9
6	19	2	19	15	15
7	29	3	29	26	26
8	47	3	47	44	45
9	71	3	79	78	80
10	127	4	127	136	141
11	191	4	191	246	256
12	379	4	379	432	455
13	607	4	635	772	822
14	1087	4	1267	1382	1499
15	1903	5	2047	2481	2746
16	3583	5	3583	4490

We have |{n ∈ N: k=m(n)=ℓ(n)-1}| = ∑_{i ≤ k} t(i)-d(i) for all k ≤ 15.

Lastly here are the numbers T(3,n), T(4,n), T(5,n), T(6,n), T(7,n), T(8,n), T(9,n), T(10,n), T(11,n), T(12,n), T(13,n), T(14,n) and T(15,n) if they are positive as plain ascii text files.