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 A304363 "Station-keeping Collatz numbers": a(n) is the smallest even number whose Collatz ('3x+1') trajectory, after its initial step downward, is directed back toward its starting value at each of its next n steps. 0
 2, 6, 6, 6, 6, 6, 22, 22, 54, 54, 118, 246, 246, 502, 502, 1014, 2038, 2038, 2038, 6134, 6134, 14326, 14326, 14326, 47094, 47094, 47094, 47094, 47094, 309238, 309238, 833526, 1882102, 1882102, 3979254, 3979254, 8173558, 8173558, 8173558, 24950774, 24950774 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS These trajectories may be called "station-keeping" in that, like a satellite correcting perturbations to its orbit as they arise, these trajectories continue (for a time) to be directed toward their starting values. (Using even numbers as starting values results in a smaller relative initial step away from the starting value, i.e., from a(n) downward to a(n)/2 rather than from a(n) upward to 3*a(n)+1.) If the Collatz conjecture is true, there are no cycles, i.e., there exists no number k > 1 whose trajectory ever returns to k, so for each starting value k, the pattern of being directed back toward k at every step after the initial downward step is eventually broken; this occurs when the trajectory reaches either an odd number greater than k (and thus moves even farther upward above k) or an even number less than k (and thus moves even farther downward below k). (For the case n=0, the trajectory from k=2, on its initial step downward, goes immediately to 1 and stops.) It can be shown that the difference between distinct consecutive terms is always a power of 2. LINKS EXAMPLE The Collatz trajectory of 6, after its initial downward step (i.e., "step 0"), is directed back toward its starting value (6) at each of the next 5 steps, but its 6th step (from 4 to 2) is directed away from the starting value: .         start of step 2    16 <-- start of step 4                |           / \ starting       v          /   \ value         10         /     \   |           / \       /       \   v          /   \     /         8 <-- start of step 5   6---------/-----\---/-----------\-----------------------    \       /       \ /             \     \     /         5               \      \   /          ^                4 <-- start of step 6       \ /           |                 \        3     start of step 3           \        ^                                \        |                                 2 start of step 1                           \                                            ... The trajectory of 2 reaches 1 (and stops) on its initial downward step, and the trajectory of 4 continues downward (to 1) after its initial downward step, so 6 is the smallest even number whose trajectory, after its initial downward step, is directed back toward its starting value for even a single step; thus a(1) = 6. Since the trajectory of 6 continues to be directed back toward its initial value for the next 4 steps as well, a(1) = a(2) = a(3) = a(4) = a(5) = 6. However, since that trajectory's 6th step after the initial downward step is directed away from the starting value, a(6) > 6. The smallest even number whose Collatz trajectory, after its initial step downward, is directed back toward its starting value at each of its next 65 steps is 1475648206838, so a(65) = 1475648206838; that trajectory continues to be directed back toward its starting value at each of the next 11 steps as well, so a(65) = a(66) = ... = a(76). PROG (Magma) nMax:=40; a:=[]; up:=0; dn:=1; kStart:=6; kAtN:=3; for n in [1..nMax] do if (3^up ge 2^dn) ne IsEven(kAtN) then kStart+:=2^dn; kAtN+:=3^up; end if; if IsEven(kAtN) then dn+:=1; kAtN:=kAtN div 2; else up+:=1; kAtN:=3*kAtN+1; end if; a[n]:=kStart; end for;  cat a; CROSSREFS Cf. A006370, A070165. Sequence in context: A184408 A184409 A137479 * A247097 A258576 A278253 Adjacent sequences:  A304360 A304361 A304362 * A304364 A304365 A304366 KEYWORD nonn AUTHOR Jon E. Schoenfield, May 11 2018 STATUS approved

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Last modified November 18 17:21 EST 2019. Contains 329287 sequences. (Running on oeis4.)