Thanks for all your interest in Charles Trigg's sequence. The current feeling is that the sequence(s) diverge sufficiently rapidly that it's likely that there are infinfinitely many non-tributary sequences, and that it's unlikely that anyone will prove anything. [has anyone found specimens of sequences which appear reluctant to join the main sequence, but which do merge themselves ??] Meanwhile, back at the ranch, there are some equal rights activists crying for equal time. The rough description of the sequence is `add the distinct prime divisors' There are four interpretations of this. 1 is not considered to be a prime, although it was not always thus (Goldbach, D.N.Lehmer,...) and it is still occasionally a useful fiction to consider 1 as the zeroth prime. So we can include or exclude 1 and, if a term is prime, we can include or exclude it. (a) we've looked at including 1 but not including the number itself, p --> p+1 (b) least interesting, perhaps is if we include neither, so, if we hit a prime, we're stuck: p --> p --> p --> ... Is there any interest ? 1,1,1,1,... 2,2,2,... 3,3,3,... 4,6,11,11,... 8,10,17,17,... 9,12,17,... 14,23,23,... 15,23,... 16,18,23,... 20,27,30,40,47,47,... 21,31,31,... 22,35,47,... 24,29,29,... 32,34,53,53,... 33,47,... 36,41,... 38,59,... 42,54,59,... 44,57,79,... E&OE What's the largest # of distinct terms that anyone can find? Perhaps here it's possible to prove something. (c) Include p but not 1: p --> 2p --> 3p+2 --> ... 2,4,6,11,22,35,47,94,143,167,334,503,... 3,6. 5,10,17,34,53,106,161,191,382,575,603,673,... 7,14,23,46,71,142,215,263,526,791,911,... 8,10. 12,17. 13,26,41,82,125,130,150,160,167,...(aha) 15,23. 16,18,23. 19,38,59,118,179,358,539,557,1114,1673,... 20,27,30,40,47. 21,31,62,95,119,143. 24,29,58,89,178,269,538,809,1618,2429,... (d) Include 1 and p (Cunningham chains!) p --> 2p+1 --> ... 1,2,5,11,23,47,95,120,131,263,527,576,582,... 3,7,15,24,30,41,83,167,335,408,431,863,... 4,7. (period, or aha! indicates tribulation) 6,12,18,24. 8,11. 9,13,27,31,63,74,114,139,279,314,474,559,... 10,18. 14,24. 16,19,39,56,66,83 (aha) 17,35,48,54,60,71,143,168,181,363,378,391, 432,438,517,576 (aha) 20,28,38,60. 21,32,35. 22,36,42,55,72,78,97,195,217,256,259,304, 326,492,539,558,595,625,631,... 25,31. 26,42. 29,59,119,144,150,161,192,198,215,264,281, 563,1127,... that's enough mistakes and cats amongst pigeons for today. Best to all, R. On Wed, 13 Apr 2005, Richard Guy wrote:
Thanks for several responses. I've got as far as Math Mag 48(1975) 301 and find:
``C.W.Trigg, C.C.Oursler, and R.Cormier & J.L.Selfridge have sent calculations on Problem 886 [Nov 1973] for which we had received only partial results [Jan 1975].
Given an ... (restatement of problem)
C & S sent the following results: There appear to be 5 seqs beginning with integers less than 1000 which do not merge. These sequences were carried out to 10^8 or more. The calculations are:
1,2,3,4,7,8,..,96532994,144799494,...(31) 393,528,545,660,682,727,...,97622612, 122028268,... (9) 412,518,565,684,709,710,..., 92029059, 102254514,... (46) 668,838,1260,1278,1355,1632,...,91127590, 100240357,... (52) 932,1168,1244,1558,1621,1622,...,98457737, 112523136,... (30)
The numbers in parens show the numbers of terms between 50000000and 10^8. The rate of growth of these sequences suggests that there are likely an inf no of mutually indep seqs.
[[30 yrs on our computers, human & electronic, shd be able to improve on this. Should the last 4 of the above 5 seqs be in OEIS ?? R.]]
On Wed, 13 Apr 2005, Richard Guy wrote:
I came across Problem 886, Math Mag 48(1975) 57--58 which isn't properly stated but should read as in OEIS A003508 :
a(n) = a(n-1) + 1 + sum of distinct prime factors of a(n-1) that are < a(n-1).