A355406 - OEIS

%I #35 Feb 04 2023 01:59:52

%S 21,42,75,84,85,113,150,151,168,170,201,226,227,267,300,301,302,336,

%T 340,341,401,402,403,423,452,453,454,475,534,535,537,600,602,604,605,

%U 633,635,672,680,682,713,715,802,803,804,805,806,846,847,891,904,906,908,909,950,951,953,955

%N Positive integers that are not powers of 2 and whose Collatz trajectory has maximum power of 2 different from 2^4.

%C It is conjectured that 15/16 (93.75%) of the positive integers that are not powers of 2 have 2^4 as the maximum power of 2 in their Collatz trajectory (see A232503 and A355187). {a(n)} lists the remaining positive integers. Consequently, it is conjectured that this sequence will have lim_{n->oo} a(n)/n = 1/16.

%C Among the numbers from 1 to 1000, there are 10 that are powers of 2, and there are 932 others (excluding 16) whose Collatz trajectories contain 2^4 as their maximum power of 2. The remaining 58 numbers are the first 58 terms of {a(n)}.

%C If k is in this sequence then so is k*2^j for any j > 0. To find a "primitive" set simply eliminate the even terms (see A350160).

%H Frank M Jackson, <a href="/A355406/b355406.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>

%e 21 is a term since its trajectory 21 64 32 16 8 2 1 has 64 as the highest power of 2, which is more than 16 and 21 is not itself a power of 2.

%t collatz[n_] := Module[{}, If[OddQ[n], 3n+1, n/2]]; step[n_] :=  Module[{p=0, m=n, q}, While[!IntegerQ[q=Log[2, m]], m=collatz[m]; p++]; {p, q, n}]; Last/@Select[Table[step[n], {n, 1, 10^5}], #[[1]]>0 && #[[2]]!=4 &]

%Y Cf. A008908, A232503, A308149, A350160, A355187.

%Y Subset of A308149 where terms that are powers of 2 have been omitted.

%K nonn

%O 1,1

%A _Frank M Jackson_, Jul 01 2022