A181777 - OEIS

%I #109 Nov 15 2024 23:35:26

%S 7,16,106,46,158,224,177,450,860,1454,1441,1660,6769,8494,17094,29821,

%T 30734,43478,55906,60716,129608,134345,153505,265860,293161,312164,

%U 598067,1158876,1482529,1771117,2906179,10197081,11568589,16927967,18807193,40055567,40663017,93778449,181209792,282515044,323346876,349304386,409093991,438465334,499902411,573966881,580260946

%N Number of steps to reach 1 in '3x+1' (or Collatz) problem starting with the n-th Mersenne prime.

%C Sequence currently gives the data for the 48 known Mersenne primes (A000043).

%C It is conjectured by Ohira and Watanabe that for large Mersenne primes 2^k-1, the fraction steps/k ~ 2+3*log(3)/log(4/3) or approximately 13.45.

%C The confirmed number of steps to reach 1 for other known Mersenne primes S(Mp) above 45th (M37156667): S(M42643801) = 573966881, S(M43112609) = 580260946, S(M57885161) = 779044992, S(M74207281) = 998401306. - Andrey S. Shchebetov and _Sergei D. Shchebetov_, Nov 14 2017

%C S(M77232917) = 1039248803. - Andrey S. Shchebetov and _Sergei D. Shchebetov_, Apr 25 2018

%C S(M82589933) = 1111148968. Also confirming all previous results. - _Martin Raab_, Apr 28 2023

%C S(M136279841) = 1833585702. - _Roderick MacPhee_, Oct 21 2024

%H Gord Palameta, <a href="/A181777/b181777.txt">Table of n, a(n) for n = 1..48</a> (first 47 terms from Andrey S. Shchebetov and Sergei D. Shchebetov)

%H Mersenneforum.org, <a href="https://www.mersenneforum.org/node/1055976?p=1057593#post1057593">Collatz "total stopping time"</a>

%H T. Ohira and H. Watanabe, <a href="http://arxiv.org/abs/1104.2804">A Conjecture on the Collatz-Kakutani Path Length for the Mersenne Primes</a>, arXiv:1104.2804 [math.NT], 2011-2012.

%H Gord Palameta, <a href="/A181777/a181777.txt">Table of a(n)/A000043(n) for n = 1..48</a>

%H Perig, <a href="https://github.com/Boutoukoat/Collatz-steps-on-large-numbers">Boutoukoat/Collatz-steps-on-large-numbers</a>

%H Martin Raab, <a href="/A181777/a181777.gp.txt">PARI program</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Collatz_conjecture">Collatz conjecture</a>

%e a(1)=7 as the first Mersenne prime is 3. So starting at 3 the steps are 10, 5, 16, 8, 4, 2, 1.

%t collatz[k_] := (If[OddQ[k], j=3k+1, j=k/2]; j); step[m_] := (p=1; n=m; While[n!=1, (n=collatz[n]; p++)]; p-1); list = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951}; Table[step[2^s-1], {s,list}] (* warning: the list should be limited so as to run in a reasonable amount of time *)

%o (PARI) \\ See Raab link. \\ _Martin Raab_, May 11 2023

%Y Cf. A000043, A006577, A070975, A193688.

%K nonn

%O 1,1

%A _Frank M Jackson_, Dec 23 2012

%E a(43)-a(45) from Andrey S. Shchebetov and _Sergei D. Shchebetov_, Sep 22 2017

%E Edited by _N. J. A. Sloane_, Sep 26 2017

%E a(46)-a(47) from _Sergei D. Shchebetov_, Apr 25 2018

%E a(48) from _Roderick MacPhee_, Oct 21 2024