A010120 -id:A010120 - cp's OEIS frontend

A288605 Position of first appearance of each integer in A088568 (number of 1's minus number of 2's in first n terms of A000002).

0, 1, 3, 12, 32, 93, 257, 378, 471, 798, 825, 858, 1127, 1398, 1497, 1524, 1533, 6352, 6969, 7176, 7269, 7566, 7971, 20338, 20371, 21982, 22009, 25638, 25665, 25692, 27969, 39184, 39211, 42398, 43129, 43150, 48637, 48730, 48757, 49014, 49041, 49068, 49095, 49864

Offset: 1

Gus Wiseman, Jun 11 2017

nonn

For the definition of this sequence we assume A088568(0) = 0.

			A088568(12) = -2 is the first appearance of -2 in A088568, so 12 belongs to the sequence.
A088568(32) = 2 is the first appearance of 2, so 32 belongs to the sequence.

Cf. A000002, A010120, A088568, A187831.

Map[First, Values@ #] - 1 &@ PositionIndex@ Prepend[#, 0] &@ MapIndexed[3 First@ #2 - 2 #1 &, Accumulate@ Prepend[Nest[Flatten[Partition[#, 2] /. {{2, 2} -> {2, 2, 1, 1}, {2, 1} -> {2, 2, 1}, {1, 2} -> {2, 1, 1}, {1, 1} -> {2, 1}}] &, {2, 2}, 25], 1]] (* Michael De Vlieger, Jun 12 2017, after Birkas Gyorgy at A000002 *)

A208981 Number of iterations required to reach a power of 2 in the 3x+1 sequence starting at n.

Original entry on oeis.org

0, 0, 3, 0, 1, 4, 12, 0, 15, 2, 10, 5, 5, 13, 13, 0, 8, 16, 16, 3, 1, 11, 11, 6, 19, 6, 107, 14, 14, 14, 102, 0, 22, 9, 9, 17, 17, 17, 30, 4, 105, 2, 25, 12, 12, 12, 100, 7, 20, 20, 20, 7, 7, 108, 108, 15, 28, 15, 28, 15, 15, 103, 103, 0, 23, 23, 23, 10, 10, 10

Offset: 1

L. Edson Jeffery, Mar 04 2012

The original name was: Number of iterations of the Collatz recursion required to reach a power of 2.
The statement that all paths must eventually reach a power of 2 is equivalent to the Collatz conjecture.
A006577(n) - a(n) gives the exponent for the first power of 2 reached in the Collatz trajectory of n. - Alonso del Arte, Mar 05 2012
Number of nonpowers of 2 in the 3x+1 sequence starting at n. - Omar E. Pol, Sep 05 2021

			a(7) = 12 because the Collatz trajectory for 7 is 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, ... which reached 16 = 2^4 in 12 steps.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
John Smith, Collatz sequence, PlanetMath.
Eric Weisstein's World of Mathematics, Collatz problem.
Index entries for sequences related to 3x+1 (or Collatz) problem

Row sums of A347519.
Cf. A006577 (and references therein).
Cf. A347270 (gives all 3x+1 sequences).
Cf. A070165, A010120, A057716, A135282, A209229.

a208981 = length . takeWhile ((== 0) . a209229) . a070165_row
-- Reinhard Zumkeller, Jan 02 2013

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

Collatz[n_?OddQ] := 3*n + 1; Collatz[n_?EvenQ] := n/2; Table[-1 + Length[NestWhileList[Collatz, n, Not[IntegerQ[Log[2, #]]] &]], {n, 50}] (* Alonso del Arte, Mar 04 2012 *)

ispow2(n)=n>>=valuation(n,2); n==1
a(n)=my(s); while(!ispow2(n), n=if(n%2, 3*n+1, n/2); s++); s \\ Charles R Greathouse IV, Jul 31 2016

For x>0 an integer, define f_0(x)=x, and for r=1,2,..., f_r(x)=f_{r-1}(x)/2 if f_{r-1}(x) is even, else f_r(x)=3*f_{r-1}(x)+1. Then a(n) = min(k such that f_k(n) is equal to a power of 2).

a(n) = A006577(n) - A135282(n) (after Alonso del Arte's comment), if A006577(n) is not -1. - Omar E. Pol, Apr 10 2022

Name clarified by Omar E. Pol, Apr 10 2022

A070167 a(n) is the smallest starting value that produces a Collatz sequence in which n occurs.

Original entry on oeis.org

1, 2, 3, 3, 3, 6, 7, 3, 9, 3, 7, 12, 7, 9, 15, 3, 7, 18, 19, 7, 21, 7, 15, 24, 25, 7, 27, 9, 19, 30, 27, 21, 33, 7, 15, 36, 37, 25, 39, 7, 27, 42, 43, 19, 45, 15, 27, 48, 43, 33, 51, 7, 15, 54, 55, 37, 57, 19, 39, 60, 27, 27, 63, 21, 43, 66, 39, 45, 69, 15, 27, 72, 73, 43, 75, 25

Offset: 1

Eric W. Weisstein, Apr 23 2002

nonn

a(n) <= n. - Robert G. Wilson v, Jan 14 2015

T. D. Noe, Table of n, a(n) for n=1..10000
Eric Weisstein's World of Mathematics, Collatz Sequence
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006667, A070165.

Cf. A010120, A054646.

import Data.List (findIndex)
import Data.Maybe (fromJust)
a070167 n = fromJust (findIndex (elem n) a070165_tabf) + 1
-- Reinhard Zumkeller, Jan 02 2013

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; nn = 100; t = Table[0, {nn}]; n = 0; zeros = nn; While[zeros > 0, n++; c = Collatz[n]; Do[If[i <= nn && t[[i]] == 0, t[[i]] = n; zeros--], {i, c}]]; t (* T. D. Noe, Dec 03 2012 *)

A054646 Smallest number to give 2^(2n) in a hailstone (or 3x + 1) sequence.

Original entry on oeis.org

1, 3, 21, 75, 151, 1365, 5461, 14563, 87381, 184111, 932067, 5592405, 13256071, 26512143, 357913941, 1431655765, 3817748707, 22906492245, 91625968981, 244335917283, 1466015503701, 5212499568715, 10424999137431

Offset: 1

Jeff Heleen, Apr 16 2000

In hailstone sequences, only even powers of 2 are obtained as a final peak before descending to 1. [I assume this should really say: "These are numbers whose 3x+1 trajectory has the property that the final peak before descending to 1 is an even power of 2." - N. J. A. Sloane, Jul 22 2020]

For n>1, this a bisection of A010120. For n=3,6,7,9,12,15,16,18,19,21, we have a(n)=(4^n-1)/3, the largest possible value because one 3x+1 step produces 2^(2n). - T. D. Noe, Feb 19 2010

			The "3x+1" sequence starting at 21 is 21, 64, 32, 16, 8, 4, 2, 1, ..., and is the smallest start which contains 64 = 2^(2*3). So a(3) = 21. - _N. J. A. Sloane_, Jul 22 2020

J. Heleen, Final Peak Sequences for Hailstone Numbers, 1993, preprint. [Apparently unpublished as of June 2017]

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

a054646 1 = 1
a054646 n = a070167 $ a000302 n  -- Reinhard Zumkeller, Jan 02 2013

For n > 1: a(n) = A070167(A000302(n)). - Reinhard Zumkeller, Jan 02 2013

A247272 Odd numbers m containing 256 as the highest power of 2 in their Collatz (3x+1) iteration.

Original entry on oeis.org

75, 85, 113, 267, 301, 401, 453, 475, 535, 633, 713, 803, 951, 1069, 1205, 1267, 1425, 1427, 1605, 1611, 1689, 1813, 1901, 2141, 2251, 2417, 2533, 2667, 2671, 2811, 2851, 2853, 3001, 3003, 3163, 3213, 3223, 3377, 3379, 3559, 3561, 3751, 3801, 3805, 3819, 3951, 4001, 4007, 4217, 4277

Offset: 1

Derek Orr, Sep 22 2014

nonn

a(n)*2^k also contains 256 as the highest power of 2 for any k >= 0.

Harvey P. Dale, Table of n, a(n) for n = 1..1000 (First 50 from Derek Orr)
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A008908, A010120.

hp256Q[n_]:=Max[Select[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&],IntegerQ[ Log[ 2,#]]&]]==256; Select[Range[1,4301,2],hp256Q] (* Harvey P. Dale, Feb 10 2019 *)

Max2(n)=v=[n];while(n!=1,if(n==Mod(0,2),n=n/2;v=concat(v,n));if(n==Mod(1,2)&&n!=1,n=3*n+1;v=concat(v,n)));k=1;while(vecsearch(vecsort(v),2^k),k++);2^(k-1)
n=1;while(n<10^4,if(n%2&&Max2(n)==256,print1(n,", "));n++)

A247346 Odd numbers n containing 1024 as the highest power of 2 in their Collatz (3x+1) iteration.

Original entry on oeis.org

151, 201, 227, 341, 403, 423, 537, 605, 635, 715, 805, 847, 891, 909, 953, 955, 1003, 1073, 1075, 1129, 1131, 1191, 1271, 1273, 1337, 1431, 1433, 1505, 1613, 1693, 1697, 1783, 1787, 1907, 1911, 2007, 2011, 2147, 2149, 2257, 2259, 2263, 2377, 2383, 2421, 2503, 2541, 2547, 2675, 2681

Offset: 1

Derek Orr, Sep 22 2014

nonn

a(n)*2^k also contains 1024 as the highest of 2 for any k >= 0.

Derek Orr, Table of n, a(n) for n = 1..50

Cf. A008908, A010120.

Max2(n)=v=[n]; while(n!=1, if(n==Mod(0, 2), n=n/2; v=concat(v, n)); if(n==Mod(1, 2)&&n!=1, n=3*n+1; v=concat(v, n))); k=1; while(vecsearch(vecsort(v), 2^k), k++); 2^(k-1)
n=1;while(n<10^4,if(n%2&&Max2(n)==1024,print1(n,", "));n++)

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.

A247715 Odd numbers n containing 16384 as the highest power of 2 in the Collatz (3x+1) iteration.

Original entry on oeis.org

5461, 7281, 29125, 38833, 51777, 77667, 103555, 116501, 122731, 138073, 155333, 163641, 184097, 207109, 245463, 276145, 276147, 310669, 327283, 368193, 368195, 414221, 414225, 436377, 466005, 490925, 552291, 552293, 581835, 621333, 654565, 689583, 736387, 736389, 828437, 828451

Offset: 1

Derek Orr, Sep 22 2014

nonn

a(n)*2^k also contains 16384 as the highest power of 2 for any k >= 0.

Cf. A008908, A010120.

Max2(n)=v=[n]; while(n!=1, if(n==Mod(0, 2), n=n/2; v=concat(v, n)); if(n==Mod(1, 2)&&n!=1, n=3*n+1; v=concat(v, n))); k=1; while(vecsearch(vecsort(v), 2^k), k++); 2^(k-1)
n=1; while(n<10^4, if(n%2&&Max2(n)==16384, print1(n, ", ")); n++)

A247716 Odd numbers n containing 65536 as the highest power of 2 in their Collatz (3x+1) iteration.

Original entry on oeis.org

14563, 17259, 19417, 20455, 21845, 25889, 27273, 30683, 34519, 38835, 40911, 46025, 51779, 54547, 58253, 61367, 64647, 69037, 72729, 77669, 81821, 92049, 92051, 96971, 103557, 107623, 109093, 109095, 122733, 129295, 138077, 143497, 145457, 145459, 153243, 155341, 161435

Offset: 1

Derek Orr, Sep 22 2014

nonn

a(n)*2^k also contains 65536 as the highest power of 2 for any k >= 0.

Cf. A008908, A010120.

Max2(n)=v=[n]; while(n!=1, if(n==Mod(0, 2), n=n/2; v=concat(v, n)); if(n==Mod(1, 2)&&n!=1, n=3*n+1; v=concat(v, n))); k=1; while(vecsearch(vecsort(v), 2^k), k++); 2^(k-1)
n=1; while(n<10^4, if(n%2&&Max2(n)==65536, print1(n, ", ")); n++)

A287319 Smallest number k which becomes a power of 2 after being transformed by the reduced Collatz function k=(3*k+1)/2 precisely n times.

Original entry on oeis.org

1, 3, 151, 26512143, 318400215865581346424671, 1240913164837493520914469575281720548839055905624577375251388717505927743

Offset: 1

Joe Slater, May 23 2017

nonn

a(7) is too large to include.

I conjecture that all members of a(n) are members of A054646 and A010120, "Smallest number to give 2^(2n) in a hailstone (3x + 1) sequence" and "Smallest start for a `3x+1' sequence containing 2^n".

			For n = 3, the reduced Collatz sequence k = (3*k+1)/2 is 151, 227, 341, 512.

Cf. A054646 and A010120.

a(n) = ((2^(3^(n-1)+n)-3^n+2^n))/3^n.

cp's OEIS Frontend

A288605 Position of first appearance of each integer in A088568 (number of 1's minus number of 2's in first n terms of A000002).

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A208981 Number of iterations required to reach a power of 2 in the 3x+1 sequence starting at n.

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A070167 a(n) is the smallest starting value that produces a Collatz sequence in which n occurs.

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A054646 Smallest number to give 2^(2n) in a hailstone (or 3x + 1) sequence.

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A247272 Odd numbers m containing 256 as the highest power of 2 in their Collatz (3x+1) iteration.

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A247346 Odd numbers n containing 1024 as the highest power of 2 in their Collatz (3x+1) iteration.

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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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A247715 Odd numbers n containing 16384 as the highest power of 2 in the Collatz (3x+1) iteration.

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