A062052 -id:A062052 - cp's OEIS frontend

A062054 Numbers with 4 odd integers in their Collatz (or 3x+1) trajectory.

17, 34, 35, 68, 69, 70, 75, 136, 138, 140, 141, 150, 151, 272, 276, 277, 280, 282, 300, 301, 302, 544, 552, 554, 560, 564, 565, 600, 602, 604, 605, 1088, 1104, 1108, 1109, 1120, 1128, 1130, 1137, 1200, 1204, 1205, 1208, 1210, 2176, 2208, 2216, 2218, 2240

Offset: 1

David W. Wilson

The Collatz (or 3x+1) function is f(x) = x/2 if x is even, 3x+1 if x is odd.
The Collatz trajectory of n is obtained by applying f repeatedly to n until 1 is reached.
A078719(a(n)) = 4; A006667(a(n)) = 3.
Numbers m such that (s0 - 4s1)/2m = 1 where s0 is the sum of the even elements and s1 the sum of the odd elements in the Collatz trajectory of m. - Michel Lagneau, Aug 13 2018
If m is in the sequence then so is 2*m, so one would only have to check odd numbers. - David A. Corneth, Aug 13 2018

			The Collatz trajectory of 17 is (17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1), which contains 4 odd integers. - _Jeffrey R. Goodwin_, Oct 26 2011

J. Shallit and D. Wilson, The "3x+1" Problem and Finite Automata, Bulletin of the EATCS #46 (1992) pp. 182-185.

David A. Corneth, Table of n, a(n) for n = 1..15549 (first 750 terms from Reinhard Zumkeller, terms < 10^15)
J. R. Goodwin, Results on the Collatz Conjecture, Sci. Ann. Comput. Sci. 13 (2003) pp. 1-16.
J. Shallit and D. Wilson, The "3x+1" Problem and Finite Automata, Bulletin of the EATCS #46 (1992) pp. 182-185.
Eric Weisstein's World of Mathematics, Collatz Problem
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for 2-automatic sequences.

Cf. A000079, A006370, A062052, A062053, A062055, A062056, A062057, A062058, A062059, A062060, A092893, A198587.

Column k=4 of A354236.

import Data.List (elemIndices)
a062054 n = a062054_list !! (n-1)
a062054_list = map (+ 1) $ elemIndices 4 a078719_list
-- Reinhard Zumkeller, Oct 08 2011

col4Q[n_]:=Module[{c=NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#!=1&]}, Count[c,?OddQ]==4]; Select[Range[2500],col4Q]  (* _Harvey P. Dale, Mar 21 2011 *)

The twelve formulas giving this sequence are listed in Corollary 3.3 in J. R. Goodwin with the following caveats: the value x cannot equal zero in formulas (3.16) and (3.20), one must multiply the formulas by all powers of 2 (2^1, 2^2, ...) to get the evens. - Jeffrey R. Goodwin, Oct 26 2011

A115774 Integers i such that 15*i = A048720bi(23,i).

Original entry on oeis.org

0, 5, 10, 20, 21, 40, 42, 80, 84, 85, 160, 168, 170, 320, 336, 340, 341, 640, 645, 672, 680, 682, 1280, 1285, 1290, 1344, 1360, 1364, 1365, 2560, 2565, 2570, 2580, 2581, 2688, 2693, 2720, 2728, 2730, 5120, 5125, 5130, 5140, 5141, 5160, 5162, 5376

Offset: 0

Antti Karttunen, Jan 30 2006

nonn

Here * stands for ordinary multiplication and A048720 is the carryless (GF(2)[X]) multiplication.

Index entries for sequences defined by congruent products between domains N and GF(2)[X]

Cf. A048717, A115767, A115770. Subset of A115772 ? A115776 gives the terms of A115772 which do not occur here. Differs from A062052 for the first time at n=18, where A115774(18)=645 while A062052(18)=672. A115775 shows this sequence in binary.

A072122 Numbers with 12 odd integers in their Collatz (or 3x+1) trajectory.

Original entry on oeis.org

39, 78, 79, 153, 156, 157, 158, 305, 306, 307, 312, 314, 315, 316, 317, 610, 611, 612, 613, 614, 624, 628, 629, 630, 631, 632, 634, 647, 683, 687, 1220, 1221, 1222, 1224, 1226, 1228, 1229, 1241, 1248, 1256, 1257, 1258, 1260, 1261, 1262, 1264, 1265, 1268

Offset: 1

Jim Nastos, Jun 19 2002

The Collatz (or 3x+1) function is f(x) = x/2 if x is even, 3x+1 if x is odd. The Collatz trajectory of n is obtained by applying f repeatedly to n until 1 is reached.

			trajectory: 39, 118, 59, 178, 89, 268, 134, 67, 202, 101, 304, 152, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1 has 12 odd numbers.

J. Shallit and D. Wilson, The "3x+1" Problem and Finite Automata, Bulletin of the EATCS #46 (1992) pp. 182-185.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
J. Shallit and D. Wilson, The "3x+1" Problem and Finite Automata, Bulletin of the EATCS #46 (1992) pp. 182-185.
Index entries for 2-automatic sequences.

Cf. A062052-A062060.

Column k=12 of A354236.

odd12Q[n_]:=Count[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&], ?OddQ]==12; Select[Range[1300],odd12Q] (* _Harvey P. Dale, Oct 17 2011 *)

A072466 Numbers with 11 odd integers in their Collatz (or 3x+1) trajectory.

Original entry on oeis.org

57, 59, 114, 115, 118, 119, 228, 229, 230, 236, 237, 238, 456, 458, 460, 461, 465, 472, 473, 474, 476, 477, 507, 513, 912, 916, 917, 920, 922, 930, 931, 943, 944, 945, 946, 947, 948, 949, 952, 954, 971, 987, 1014, 1015, 1025, 1026, 1027, 1031, 1129, 1131

Offset: 1

Jim Nastos, Jun 19 2002

The Collatz (or 3x+1) function is f(x) = x/2 if x is even, 3x+1 if x is odd. The Collatz trajectory of n is obtained by applying f repeatedly to n until 1 is reached.

J. Shallit and D. Wilson, The "3x+1" Problem and Finite Automata, Bulletin of the EATCS #46 (1992) pp. 182-185.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
J. Shallit and D. Wilson, The "3x+1" Problem and Finite Automata, Bulletin of the EATCS #46 (1992) pp. 182-185.
Index entries for 2-automatic sequences.

Cf. A062052-A062060.

Column k=11 of A354236.

b:= proc(n) option remember; irem(n, 2, 'r')+
      `if`(n=1, 0, b(`if`(n::odd, 3*n+1, r)))
    end:
q:= n-> is(b(n)=11):
select(q, [$1..2000])[];  # Alois P. Heinz, May 18 2022

ocollQ[n_]:=Length[Select[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&],OddQ[#]&]]==11; Select[Range[1140],ocollQ[#]&] (* Jayanta Basu, May 28 2013 *)

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A062054 Numbers with 4 odd integers in their Collatz (or 3x+1) trajectory.

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A115774 Integers i such that 15*i = A048720bi(23,i).

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A072122 Numbers with 12 odd integers in their Collatz (or 3x+1) trajectory.

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A072466 Numbers with 11 odd integers in their Collatz (or 3x+1) trajectory.

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