I'm collecting Murray Klamkin problems and solutions and am currently going thru Math Mag. I came across Problem 886, Math Mag 48(1975) 57--58 [nothing to do with Murray] 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). This leads to 1,2,3,4,7,8,11,12,18,24,30,41,42,55,... The original problem asked that if you start elsewhere, e.g., 5,6,12, ... or 9,13,14,24, ... or 10,18, ... or 15,24, ... do you always merge with the original sequence? Evidently 91,112,122,186,... takes a little while. Has anyone ... Can anyone prove Charles Trigg's guess ? R.
do you always merge with the original sequence? Evidently
91,112,122,186,... takes a little while.
Has anyone ... Can anyone prove Charles Trigg's guess ? R.
Beats me. 91 rejoins the sequence at 685 in 23 steps. 105 rejoins the sequence at 3763 in 40 steps. -- Chuck
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'm collecting Murray Klamkin problems and solutions and am currently going thru Math Mag.
I came across Problem 886, Math Mag 48(1975) 57--58 [nothing to do with Murray] 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).
This leads to 1,2,3,4,7,8,11,12,18,24,30,41,42,55,...
The original problem asked that if you start elsewhere, e.g.,
5,6,12, ... or 9,13,14,24, ... or 10,18, ... or 15,24, ...
do you always merge with the original sequence? Evidently
91,112,122,186,... takes a little while.
Has anyone ... Can anyone prove Charles Trigg's guess ? R.
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Hello, 14 avr 2005, 01:23 Europe/Brussels, Richard Guy wrote :
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)
(...)
Should the last 4 of the above 5 seqs be in OEIS ?? R.]]
I would suggest also the sequence of the beginnings of the sequences which do not merge, for instance here (to be completed!): 393, 412, 668, 932... Alexandre
[[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.]]
The sequence beginning with 1 (call it s1) and the sequence beginning with 393 (call it s2) appear to have no terms in common in their first 5000 terms. The 5000'th term of s1 is 425206306029958511853562109954. The 5000'th term of s2 is 18795034909653422153407847455617. -- Chuck
rkg:
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.
Dumb heuristics and brief numerical experimentation point to the sequences growing as around e^sqrt(n). For random sequences of about that rate of growth, you'd expect them to intersect when they are small if at all; once they get over, say, e^10 (about 22000), there would be about a 2% chance of an intersection. But it would take more thinking than I have available to make a guess at the rate of growth of the minimal starting terms for independent ones... --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.
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).
participants (5)
-
alexandre.wajnberg＠skynet.be -
Chuck Seggelin -
Chuck Seggelin -
Michael Kleber -
Richard Guy