A160541 - OEIS

%I #53 Jul 11 2023 12:04:48

%S 0,1,1,1,1,2,3,1,4,2,3,2,2,4,2,1,3,5,4,2,1,4,2,2,5,3,17,4,4,3,16,1,6,

%T 4,2,5,4,5,6,2,17,2,6,4,4,3,16,2,5,6,5,3,2,18,17,4,7,5,6,3,3,17,15,1,

%U 6,7,5,4,3,3,16,5,18,5,2,5,5,7,6,2,4,18,17

%N Number of odd-then-even runs to reach 1 from n under the modified "3x+1" map: x -> x/2 if x is even, x -> (3x+1)/2 if x is odd.

%C The 2->1 step is not counted.

%C From _Dustin Theriault_, May 24 2023: (Start)

%C The ratio of the partial sum of a(n) to the partial sum of A006577(n) appears to approach 1/6 (observation for n = 1..10^10).

%C The ratio of the partial sum of a(n) to the partial sum of A286380(n) appears to approach 1/2 (observation for n = 1..10^10). (End)

%H Michael De Vlieger, <a href="/A160541/b160541.txt">Table of n, a(n) for n = 1..10000</a>

%H Francis Laclé, <a href="https://hal.archives-ouvertes.fr/hal-03201180v2">2-adic parity explorations of the 3n+1 problem</a>, hal-03201180v2 [cs.DM], 2021.

%H Dustin Theriault, <a href="/A160541/a160541.png">Ratio between the partial sum of a(n) to the partial sum of A006577(n), for n = 1..10^10</a>.

%H Dustin Theriault, <a href="/A160541/a160541_2.png">Ratio between the partial sum of  a(n) to the partial sum of A286380(n), for n = 1..10^10</a>.

%H Dustin Theriault, <a href="/A160541/a160541_1.png">Histogram of a(n), for n = 1..10^10</a>.

%H Dustin Theriault, <a href="/A160541/a160541_3.png">Combined histograms of A006577, A286380, A160541, for n = 1..10^9</a>.

%F Number of steps x -> A363270(x) to go from n to 1. - _Dustin Theriault_, Jul 09 2023

%e 7->11->17->26->13->20->10->5->8->4->2->1, so the odd-then-even runs are (7->11->17->26) (13->20->10) (5->8->4->2), and a(7) is 3.

%t Array[Length@ Split[Most@ NestWhileList[If[EvenQ@ #, #/2, (3 # + 1)/2] &, #, # > 1 &], Or[OddQ[#1], EvenQ[#2]] &] &, 120] (* Corrected by _Michael De Vlieger_, Jul 19 2021 *)

%o (C) int a(int n) {

%o   int steps = 0;

%o   while (n > 1) {

%o     while (n & 1) n += (n >> 1) + 1;

%o     while (!(n & 1)) n >>= 1;

%o     ++steps;

%o   }

%o   return steps;

%o } /* _Dustin Theriault_, May 23 2023 */

%Y Cf. A006577, A286380, A363270.

%K nonn

%O 1,6

%A Brenton Bostick (bostick(AT)gmail.com), May 18 2009