A005184 Self-contained numbers: odd numbers k whose Collatz sequence contains a higher multiple of k. (Formerly M5220)

31, 83, 293, 347, 671, 19151, 2025797

Offset

1,1

Comments

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

References

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).

Links

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).

Example

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".

Mathematica

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 *)

Prog

(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)

Crossrefs

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

Keyword

nonn,more

Author

N. J. A. Sloane

Extensions

More terms from Robert G. Wilson v

Better description from Jack Brennen, Feb 07 2003

Status

approved