When Eric said minimal, did he mean that a[n+1]-a[n] had to be positive? If you look at the non-monotonically increasing version of the sequence, and compare it to the monotonic version, then the two sequences diverge after 1002. There are only 91 0's at this point, so the next term after 1002 in the non-monotonic sequence would be 920 instead of 1003. Some interesting things turn up if you compare the monotonic and non-monotonic versions of the sequence in different bases. In base 2 the sequence doesn't change whether you require monotonicity or not. However, in base 3 the monotonic sequence is only 10 terms, but the non-monotonic sequence goes for 59 terms (it diverges after the 9th term). The monotonic sequence ends: .... 1021 1121. The non-monotonic sequence goes: .... 1021 210 220 ... In base 4, the monotonic sequence is 22 terms, and the non-monotonic sequence diverges, again, at the 2nd to last term, and goes on for a total of 41 terms. Interestingly, the last non-divergent term is 1002 (base 4), which is the same term that base 10 diverged at. Monotonic: ... 1002 1012. non-monotonic: ... 1002 320 330 ... Base 5 also diverges after 1002 (base 5)! The monotonic sequence is 61 elements long, the non-monotonic seq is 2658 terms long. They diverge after 43 terms: Monotonic: ... 1002 1012 1013 ... non-monotonic: ... 1002 420 430 ... Base 6 also diverges after 1002 (base 6)! The monotonic sequence is 248 elements long, the non-monotonic seq is 4468 terms long. They diverge after 75 terms: Monotonic: ... 1002 1004 1012 1013 ... non-monotonic: ... 1002 520 530 540 ... The pattern breaks at base 7, where it diverges at 1004 --- 1002 doesn't appear in the sequence monotonic sequence. It resumes for bases 8, 9, and 10, though. 11 breaks the pattern again, diverging after 4005 (base 11), followed by 4006 (base 11) in the monotonic sequence, and by 3A20 in the non-monotonic sequence. Tue, 22 Feb 2005 09:17:40 -0500 "David Wilson" <davidwwilson@comcast.net> Because my results disagreed with Chuck Seggelin's, I rechecked my program. There was a subtle bug in the termination code that caused me to drop the last three elements. When fixed, my sequence coincides with Seggelin's. The finity argument I gave before is basically sound. At a(2024) = 8945, the largest digit count is 894 5's. There are no more elements up to 8950. At 8950 and beyond, the "count" part of the number exceeds 894, and so cannot be in the sequence. Ergo the sequence is finite, with largest element a(2024) = 8945. -- No virus found in this outgoing message. Checked by AVG Anti-Virus. Version: 7.0.300 / Virus Database: 266.2.0 - Release Date: 2/21/2005 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun