A355406 Positive integers that are not powers of 2 and whose Collatz trajectory has maximum power of 2 different from 2^4.

21, 42, 75, 84, 85, 113, 150, 151, 168, 170, 201, 226, 227, 267, 300, 301, 302, 336, 340, 341, 401, 402, 403, 423, 452, 453, 454, 475, 534, 535, 537, 600, 602, 604, 605, 633, 635, 672, 680, 682, 713, 715, 802, 803, 804, 805, 806, 846, 847, 891, 904, 906, 908, 909, 950, 951, 953, 955

Offset

1,1

Comments

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.

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)}.

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).

Links

Frank M Jackson, Table of n, a(n) for n = 1..10000

Index entries for sequences related to 3x+1 (or Collatz) problem

Example

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.

Mathematica

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 &]

Crossrefs

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

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

Sequence in context: A086794 A001682 A180963 * A078440 A175805 A039344

Adjacent sequences: A355403 A355404 A355405 * A355407 A355408 A355409

Keyword

nonn

Author

Frank M Jackson, Jul 01 2022

Status

approved