A181777 Number of steps to reach 1 in '3x+1' (or Collatz) problem starting with the n-th Mersenne prime.

7, 16, 106, 46, 158, 224, 177, 450, 860, 1454, 1441, 1660, 6769, 8494, 17094, 29821, 30734, 43478, 55906, 60716, 129608, 134345, 153505, 265860, 293161, 312164, 598067, 1158876, 1482529, 1771117, 2906179, 10197081, 11568589, 16927967, 18807193, 40055567, 40663017, 93778449, 181209792, 282515044, 323346876, 349304386, 409093991, 438465334, 499902411, 573966881, 580260946

Offset: 1

Author

Frank M Jackson, Dec 23 2012

Keywords

• nonn

Comments

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

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.

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

S(M77232917) = 1039248803. - Andrey S. Shchebetov and Sergei D. Shchebetov, Apr 25 2018

S(M82589933) = 1111148968. Also confirming all previous results. - Martin Raab, Apr 28 2023

S(M136279841) = 1833585702. - Roderick MacPhee, Oct 21 2024

Examples

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

Links

• Gord Palameta, Table of n, a(n) for n = 1..48 (first 47 terms from Andrey S. Shchebetov and Sergei D. Shchebetov)
• Mersenneforum.org, Collatz "total stopping time"
• T. Ohira and H. Watanabe, A Conjecture on the Collatz-Kakutani Path Length for the Mersenne Primes, arXiv:1104.2804 [math.NT], 2011-2012.
• Gord Palameta, Table of a(n)/A000043(n) for n = 1..48
• Perig, Boutoukoat/Collatz-steps-on-large-numbers
• Martin Raab, PARI program
• Wikipedia, Collatz conjecture

Crossrefs

Cf. A000043, A006577, A070975, A193688.

Programs

Mathematica

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 *)

PARI

\\ See Raab link. \\ Martin Raab, May 11 2023

Extensions

a(43)-a(45) from Andrey S. Shchebetov and Sergei D. Shchebetov, Sep 22 2017

Edited by N. J. A. Sloane, Sep 26 2017

a(46)-a(47) from Sergei D. Shchebetov, Apr 25 2018

a(48) from Roderick MacPhee, Oct 21 2024