Closures and Complements

[Don Reble]

> An interesting thing happens when we look at the closure of {2} under
> ab-1 and ab+1. The resulting sequence is the complement of A014574
> (midpoints of twin primes), except for the absence of 1. (This is not
> hard to prove.)
    Kewl! But please tell us whether the complement is finite. :-)

> The next sequence that caught my attention is the closure of {2} under
> ab and ab+1. This produces a sequence whose complement is finite. The
> complete complement is below:
> 
> 1 3 6 7 8...
> This was calculated by hand, so it may contain errors.

Indeed, "8" is in the closure. I get

1 3 6 7 12 13 14 15 24 27 28 29 30 31 48 49 54 57 58 59 60 61 62 63 97
98 108 109 114 117 118 119 120 123 124 127 194 197 217 218 219 228 229
237 238 239 240 241 246 247 248 249 389 394 434 439 457 458 479 482 492
493 497 498 499 788 789 878 879 917 959 984 985 986 994 997 998 1579
1757 1758 1759 1968 1971 1988 1994 1997 3514 3517 3518 3989 7028 7034
7037 7978 14074 28148

And this time, finiteness is proven easily.

--
Don Reble       djr at nk.ca