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A301937
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a(n) is the smallest number whose Collatz ('3x+1') trajectory crosses its initial value exactly n times.
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2
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1, 3, 10, 7, 6, 9, 22, 19, 14, 25, 18, 83, 62, 33, 54, 559, 108, 109, 110, 97, 188, 147, 166, 221, 146, 171, 292, 129, 194, 257, 294, 313, 342, 399, 506, 609, 462, 353, 398, 531, 834, 471, 530, 1153, 9854, 417, 470, 627, 8758, 9853, 626, 9225, 18450, 20609, 23718
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OFFSET
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0,2
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COMMENTS
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Records: 1, 3, 10, 22, 25, 83, 559, 609, 834, 1153, 9854, 18450, 20609, 23718, 31142, 35090, 41586, 80294, 283262, 377681, 427762, 789305, 887954, 887964, 1403202, 1752022, ..., . - Robert G. Wilson v, May 06 2018
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LINKS
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FORMULA
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If the Collatz conjecture is true, then a(n) == n (mod 2) for all terms.
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EXAMPLE
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The Collatz trajectory for k=3 is 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 crosses the threshold value of 3 on exactly one iteration: the iteration on which it moves from 4 to 2. No smaller value of k shares this property, so a(1) = 3.
The Collatz trajectory for k=6 (see A304030) is nearly identical, containing, in order of appearance, the values 6, 3, 10, 5, 16, 8, 4, 2, 1; it crosses the threshold value of 6 on exactly 4 iterations (3 -> 10, 10 -> 5, 5 -> 16, and 8 -> 4). No smaller value of k shares this property, so a(4) = 6.
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MATHEMATICA
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Collatz[n_] := NestWhileList[ If[ OddQ@#, 3# +1, #/2] &, n, # > 1 &]; f[n_] := Block[{x = Length[ SplitBy[ Collatz@ n, # < n +1 &]] - 1}, If[ OddQ@ n && n > 1, x - 1, x]]; t[_] := 0; k = 1; While[k < 24000, If[ t[f[k]] == 0, t[f[k]] = k]; k++]; t@# & /@ Range@54 (* Robert G. Wilson v, May 05 2018 *)
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PROG
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(Magma) nMax:=54; a:=[0: n in [1..nMax]]; for k in [2..24000] do n:=0; t:=k; while t gt 1 do tPrev:=t; if IsEven(t) then t:=t div 2; else t:=3*t+1; end if; if (t-k)*(tPrev-k) lt 0 then n+:=1; end if; end while; if (n gt 0) and (n le nMax) then if a[n] eq 0 then a[n]:=k; end if; end if; end for; a;
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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