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A005184
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Self-contained numbers: odd numbers k whose Collatz sequence contains a higher multiple of k.
(Formerly M5220)
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3
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OFFSET
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1,1
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COMMENTS
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The Collatz sequence of a number k is defined as a(1)=k, a(j+1) = a(j)/2 if a(j) is even, 3*a(j) + 1 if a(j) is odd.
No others less than 250000000. - Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 07 2006
There are no more terms < 10^11. - Donovan Johnson, Nov 28 2013
There are no more terms < 10^15. - Alun Stokes, Mar 01 2021
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, E16.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Table of n, a(n) for n=1..7.
Alexander Rahn, Max Henkel, Sourangshu Ghosh, Eldar Sultanow, and Idriss Aberkane, An algorithm for linearizing Collatz convergence, hal-03286608 [math.DS], 2021.
Eldar Sultanow, Christian Koch, and Sean Cox, Collatz Sequences in the Light of Graph Theory, Universität Potsdam (Germany, 2020).
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EXAMPLE
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The Collatz sequence of 31 is 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310 (see A008884) ... 310 is a multiple of 31, so the number 31 is "self-contained".
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MATHEMATICA
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isSelfContained[n_] := Module[{d}, d = n; While[d != 1, If[EvenQ[d], d = d/2, d = 3 * d + 1]; If[IntegerQ[d/n], Return[True]]]; Return[False]]; For[n = 1, n <= 250000000, n += 2, If[isSelfContained[n], Print[n]]]; (* Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 07 2006 *)
scnQ[n_] := MemberQ[Divisible[#, n] & / @Rest[NestWhileList[If[EvenQ[#], #/2, 3# + 1] &, n, # > 1 &]], True]; Select[Range[1, 2100001, 2], scnQ] (* Harvey P. Dale, Oct 21 2011 *)
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PROG
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(PARI) m=5; d=2; while(1, n=(3*m+1)\2; until(n==1, n=if(n%2, 3*n+1, n\2); if(n%m==0, print(m, " ", n); break)); m+=d; d=6-d)
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CROSSREFS
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The ratios "higher multiple of k" / k are given in A059198.
Sequence in context: A044550 A055810 A142522 * A096731 A039518 A179113
Adjacent sequences: A005181 A005182 A005183 * A005185 A005186 A005187
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KEYWORD
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nonn,more
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Robert G. Wilson v
Better description from Jack Brennen, Feb 07 2003
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STATUS
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approved
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