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A067433
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Triangle in which row n gives trajectory of n under the map k -> k/3 if k is divisible by 3, otherwise k -> next multiple of 3, stopping when reaching 1 (the initial term n is not included).
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1
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1, 3, 1, 1, 6, 2, 3, 1, 6, 2, 3, 1, 2, 3, 1, 9, 3, 1, 9, 3, 1, 3, 1, 12, 4, 6, 2, 3, 1, 12, 4, 6, 2, 3, 1, 4, 6, 2, 3, 1, 15, 5, 6, 2, 3, 1, 15, 5, 6, 2, 3, 1, 5, 6, 2, 3, 1, 18, 6, 2, 3, 1, 18, 6, 2, 3, 1, 6, 2, 3, 1, 21, 7, 9, 3, 1, 21, 7, 9, 3, 1, 7, 9, 3
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OFFSET
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1,2
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COMMENTS
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These numbers converge to various last 3-digit endings and only to 2 last 2-digit numbers: 2,1 or 3,1. m=3. p=1 below. If m=2, p=1 you get the x+1 conjecture. If m=2, p=3 you get the 3x+1 conjecture. See Link for numbers with a large number of digits. Other conjectures are possible by trial-and-error input of m and n. It is interesting to note that for many m and p=m+1 the program converges to 1. However, for m prime and p=m+1 the program always converges to m^2, m, 1. Also for m+1 prime the program converges to m^2, m, 1 most of the time. An exception is m=6. The sequence converges but to what I call an uninteresting ending.
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LINKS
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EXAMPLE
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4 -> 6 -> 2 -> 3 -> 1, so row 4 is 6,2,3,1. Row 5 is the same.
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MATHEMATICA
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nxt[n_]:=If[Divisible[n, 3], n/3, 3(Floor[n/3]+1)]; Join[{1}, Flatten[ Table[ Rest[ NestWhileList[nxt, i, #!=1&]], {i, 30}]]] (* Harvey P. Dale, Sep 16 2012 *)
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PROG
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(PARI) multxp2(n, m, p) = { print1(1" "); for(j=1, n, x=j; c=0; while(x>1, r = x%m; if(r==0, x=x/m, x=x*p+m-r); print1(x" "); ); ) }
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CROSSREFS
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KEYWORD
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easy,tabf,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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