Closures and Complements

Sat May 22 02:08:52 CEST 2004

I have been looking recently at sequences generated by starting with some set of integers, and closing the set under an operation - in particular, under multiplication and adding a value from a predetermined list of constants.  I have located two such sequences on the OEIS: A009388, which is the closure of {2} under ab-1, and A009293, the closure of {2} under ab+1.

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.)

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 12 13 14 15 23 24 27 28 29 30 31 47 48 49 54 57 58 59 60 61 62 63 94 97 98 108 109 114 115 116 117 118 119 120 123 124 127 188 194 197 217 218 219 228 229 233 236 237 238 239 240 241 246 247 248 249 254 376 377 389 394 434 435 436 437 438 439 457 458 466 467 472 479 492 493 497 498 499 508 509 752 753 754 778 788 789 871 872 873 876 877 878 879 917 932 933 934 944 959 984 985 986 987 994 997 998 1019 1504 1509 1556 1579 1741 1747 1752 1757 1758 1759 1834 1867 1868 1869 1889 1968 1971 1972 1973 1974 1988 1994 1997 2038 2039 3008 3018 3019 3482 3494 3504 3514 3517 3518 3669 3734 3779 3942 3943 3946 3947 3948 3989 4079 6037 6038 6989 7028 7034 7037 7559 7884 7892 7893 7894 7895 7978 8158 12074 12077 13979 14074 15118 15119 15784 15788 15957 27958 28148 30237 31568 31914.

This was calculated by hand, so it may contain errors.  I would appreciate it if someone could verify it.

This generalizes to the closure of {n} under ab+k, where 0<=k<n.  I conjecture that the complement of this closure is finite for any n > 1, although I have not actually verified it for any n > 2.  If this conjecture is correct, it suggests a couple of more sequences: the maximum value in the complement for each n, and the number of values in the complement.

-- 
Franklin T. Adams-Watters
16 W. Michigan Ave.
Palatine, IL 60067
847-776-7645

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