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%I M5220 #51 Nov 24 2021 03:04:11
%S 31,83,293,347,671,19151,2025797
%N Self-contained numbers: odd numbers k whose Collatz sequence contains a higher multiple of k.
%C 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.
%C No others less than 250000000. - Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 07 2006
%C There are no more terms < 10^11. - _Donovan Johnson_, Nov 28 2013
%C There are no more terms < 10^15. - _Alun Stokes_, Mar 01 2021
%D R. K. Guy, Unsolved Problems in Number Theory, E16.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alexander Rahn, Max Henkel, Sourangshu Ghosh, Eldar Sultanow, and Idriss Aberkane, <a href="https://hal.archives-ouvertes.fr/hal-03286608">An algorithm for linearizing Collatz convergence</a>, hal-03286608 [math.DS], 2021.
%H Eldar Sultanow, Christian Koch, and Sean Cox, <a href="https://doi.org/10.25932/publishup-48214">Collatz Sequences in the Light of Graph Theory</a>, Universität Potsdam (Germany, 2020).
%e 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".
%t 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 *)
%t 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 *)
%o (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)
%Y The ratios "higher multiple of k" / k are given in A059198.
%K nonn,more
%O 1,1
%A _N. J. A. Sloane_
%E More terms from _Robert G. Wilson v_
%E Better description from _Jack Brennen_, Feb 07 2003
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