A232503 -id:A232503 - cp's OEIS frontend

A135282 Largest k such that 2^k appears in the trajectory of the Collatz 3x+1 sequence started at n.

0, 1, 4, 2, 4, 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 8, 4, 4, 4, 4, 4, 4, 4, 4, 6, 8, 4, 4

Offset: 1

Masahiko Shin, Dec 02 2007

nonn

Most of the first eighty terms in the sequence are 4, because the trajectories finish with 16 -> 8 -> 4 -> 2 -> 1. - R. J. Mathar, Dec 12 2007
Most of the first ten thousand terms are 4, and there only appear 4, 6, 8, and 10 in the extent, unless n is power of 2. In the other words, it seems that the trajectory of the Collatz 3x + 1 sequence ends with either 16, 64, 256 or 1024. There are few exceptional terms, for example a(10920) = 12, a(10922) = 14. It also seems all terms are even unless n is an odd power of 2. - Masahiko Shin, Mar 16 2010
It is true that all terms are even unless n is an odd power of 2: 2 == -1 mod 3, 2 * 2 == -1 * -1 == 1 mod 3. Therefore only even-indexed powers of 2 are congruent to 1 mod 3 and thus reachable by either a halving step or a "tripling step," whereas the odd-indexed powers of 2 are only reachable by a halving step or as an initial value. - Alonso del Arte, Aug 15 2010

			a(6) = 4 because the sequence is 6, 3, 10, 5, 16, 8, 4, 2, 1; there 16 = 2^4 is the largest power of 2 encountered.

Cf. A007814, A209229, A070165, A232503.

Cf. A006577, A208981.

#include  int main(){ int i, s, f; for(i = 2; i < 10000; i++){ f = 0; s = i; while(s != 1){ if(s % 2 == 0){ s = s/2; f++;} else{ f = 0; s = 3 * s + 1; } } printf("%d,", f); } return 0; } /* Masahiko Shin, Mar 16 2010 */

a135282 = a007814 . head . filter ((== 1) . a209229) . a070165_row
-- Reinhard Zumkeller, Jan 02 2013

A135282 := proc(n) local k,threen1 ; k := 0 : threen1 := n ; while threen1 > 1 do if 2^ilog[2](threen1) = threen1 then k := max(k,ilog[2](threen1)) ; fi ; if threen1 mod 2 = 0 then threen1 := threen1/2 ; else threen1 := 3*threen1+1 ; fi ; od: RETURN(k) ; end: for n from 1 to 80 do printf("%d, ",A135282(n)) ; od: # R. J. Mathar, Dec 12 2007

Collatz[n_] := If[EvenQ[n], n/2, 3*n + 1]; Log[2, Table[NestWhile[Collatz, n, ! IntegerQ[Log[2, #]] &], {n, 100}]] (* T. D. Noe, Mar 05 2012 *)

a(n) = A006577(n) - A208981(n) (after Alonso del Arte's comment in A208981), if A006577(n) is not -1. - Omar E. Pol, Apr 10 2022

Edited and extended by R. J. Mathar, Dec 12 2007

More terms from Masahiko Shin, Mar 16 2010

A347532 a(n) is the sum of the nonpowers of 2 in the 3x+1 sequence that starts at n.

Original entry on oeis.org

0, 0, 18, 0, 5, 24, 257, 0, 308, 15, 228, 36, 88, 271, 663, 0, 183, 326, 488, 35, 21, 250, 602, 60, 627, 114, 101409, 299, 411, 693, 101073, 0, 810, 217, 509, 362, 504, 526, 2313, 75, 101300, 63, 1307, 294, 466, 648, 100948, 108, 775, 677, 1099, 166, 368, 101463, 102285, 355

Offset: 1

Omar E. Pol, Sep 05 2021

nonn

a(n) is the sum of the nonpowers of 2 in the n-th row of A347270.

a(n) = 0 if and only if n is a power of 2.

			For n = 6 the 3x+1 sequence starting at 6 is 6, 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, ... Only the first four terms are nonpowers of 2. The sum of them is 6 + 3 + 10 + 5 = 24, so a(6) = 24.

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

Cf. A208981 (number of nonpowers of 2).

Cf. A000079, A006370, A033493, A033495, A057716, A135282, A232503, A347270, A347519.

a:= proc(n) option remember; `if`(n=2^ilog2(n), 0,
      n+a(`if`(n::odd, 3*n+1, n/2)))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 05 2021

a[n_] := Plus @@ Select[NestWhileList[If[OddQ[#], 3*# + 1, #/2] &, n, # > 1 &], # != 2^IntegerExponent[#, 2] &]; Array[a, 50] (* Amiram Eldar, Sep 06 2021 *)

From Alois P. Heinz, Sep 05 2021: (Start)
a(n) = A033493(n) - 2 * A232503(n) + 1.
a(n) = A033493(n) - 2^(A135282(n)+1) + 1. (End)

A231610 The least k such that the Collatz (3x+1) iteration of k contains 2^n as the largest power of 2.

Original entry on oeis.org

1, 2, 4, 8, 3, 32, 21, 128, 75, 512, 151, 2048, 1365, 8192, 5461, 32768, 14563, 131072, 87381, 524288, 184111, 2097152, 932067, 8388608, 5592405, 33554432, 13256071, 134217728, 26512143, 536870912, 357913941, 2147483648, 1431655765, 8589934592, 3817748707

Offset: 0

T. D. Noe, Dec 02 2013

nonn

Very similar to A225124, where 2^n is the largest number in the Collatz iteration of A225124(n). The only difference appears to be a(8), which is 75 here and 85 in A225124. The Collatz iteration of 75 is {75, 226, 113, 340, 170, 85, 256, 128, 64, 32, 16, 8, 4, 2, 1}.

			The iteration for 21 is {21, 64, 32, 16, 8, 4, 2, 1}, which shows that 64 = 2^6 is a term. However, 32 is not the first power of two. We have to wait until the iteration for 32, which is {32, 16, 8, 4, 2, 1}, to see 32 = 2^5 as the first power of two.

Cf. A010120, A054646 (similar sequences).
Cf. A135282, A232503 (largest power of 2 in the Collatz iteration of n).
Cf. A225124.

Collatz[n_?OddQ] := 3*n + 1; Collatz[n_?EvenQ] := n/2; nn = 21; t = Table[-1, {nn}]; n = 0; cnt = 0; While[cnt < nn, n++; q = Log[2, NestWhile[Collatz, n, Not[IntegerQ[Log[2, #]]] &]]; If[q < nn && t[[q + 1]] == -1, t[[q + 1]] = n; cnt++]]; t

a(n) = 2^n for odd n.

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

Original entry on oeis.org

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

Frank M Jackson, Jul 01 2022

nonn

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).

			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.

Frank M Jackson, Table of n, a(n) for n = 1..10000
Index entries for sequences related to 3x+1 (or Collatz) problem

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

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

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 &]

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A135282 Largest k such that 2^k appears in the trajectory of the Collatz 3x+1 sequence started at n.

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A347532 a(n) is the sum of the nonpowers of 2 in the 3x+1 sequence that starts at n.

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A231610 The least k such that the Collatz (3x+1) iteration of k contains 2^n as the largest power of 2.

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

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