

A301937


a(n) is the smallest number whose Collatz ('3x+1') trajectory crosses its initial value exactly n times.


2



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

0,2


COMMENTS

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


LINKS

Jon E. Schoenfield, Table of n, a(n) for n = 0..185
Index entries for sequences related to 3x+1 (or Collatz) problem


FORMULA

a(n) = min{k : A304030(k) = n}.


EXAMPLE

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.
If the Collatz conjecture is true, then a(n) == n (mod 2) for all terms.


MATHEMATICA

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 *)


PROG

(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 (tk)*(tPrevk) 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;


CROSSREFS

Cf. A006370, A070165, A304030.
Sequence in context: A196163 A195922 A261836 * A185139 A300786 A182241
Adjacent sequences: A301934 A301935 A301936 * A301938 A301939 A301940


KEYWORD

nonn


AUTHOR

Jon E. Schoenfield, May 05 2018


STATUS

approved



