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A303698
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Numbers k whose trajectory in the Collatz (or '3x+1') problem includes another multiple of k.
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0
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31, 62, 83, 166, 293, 347, 586, 671, 694, 1342, 2684, 19151, 38302, 2025797, 4051594
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OFFSET
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1,1
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COMMENTS
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If a term k is even, then k/2 is also in the sequence.
For each odd term k among the first 15 terms, 2*k is also a term. Does the sequence include any odd terms k for which 2*k is not also in the sequence?
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LINKS
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EXAMPLE
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The Collatz trajectory of 31 begins 31 -> 94 -> 47 -> 142 -> 71 -> 214 -> 107 -> 322 -> 161 -> 484 -> 242 -> 121 -> 364 -> 182 -> 91 -> 274 -> 137 -> 412 -> 206 -> 103 -> 310 -> 155 -> 466 -> ... which contains 310 and 155, both of which are multiples of 31, so 31 is in the sequence.
Other than its initial value, the trajectory of 62 is the same as that of 31, so it also includes 310, which is a multiple of 62, so 62 is in the sequence.
The trajectory of 671 includes 29524 = 671 * 11 * 2^2, so the sequence includes 671, 671*2 = 1342, and 671*4 = 2684.
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MATHEMATICA
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traj[1]:={1};
traj[n_]:=traj[n]=If[EvenQ[n]&&n>0, {n}~Join~traj[n/2], {n}~Join~traj[3*n+1]];
fQ[n_]:=Select[traj[n], IntegerQ[#/n]&&#/n>1&, 1]!={};
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PROG
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(Magma) a:=[]; for k in [2..40000] do t:=k; while t gt 1 do if IsEven(t) then t:=t div 2; else t:=3*t+1; end if; if IsDivisibleBy(t, k) then a[#a+1]:=k; break; end if; end while; end for; a; // Jon E. Schoenfield, Apr 30 2018
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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