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

1,1

Comments

Sequence currently gives the data for the 47 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

Links

Andrey S. Shchebetov and Sergei D. Shchebetov, Table of n, a(n) for n = 1..47

T. Ohira and H. Watanabe, A Conjecture on the Collatz-Kakutani Path Length for the Mersenne Primes, arXiv:1104.2804 [math.NT], 2011-2012.

Martin Raab, PARI program

Wikipedia, Collatz conjecture

Example

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

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 *)

Prog

(PARI) \\ See Raab link. \\ Martin Raab, May 11 2023

Crossrefs

Cf. A000043, A006577, A070975, A193688.

Sequence in context: A214904 A029498 A351531 * A332573 A129422 A184062

Adjacent sequences: A181774 A181775 A181776 * A181778 A181779 A181780

Keyword

nonn

Author

Frank M Jackson, Dec 23 2012

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

Status

approved