A198587 -id:A198587 - cp's OEIS frontend

A385110 Terms of A198587 congruent {1, 3, 7} (mod 8).

17, 35, 75, 151, 1137, 2275, 2417, 4835, 4849, 9699, 19417, 38833, 38835, 72817, 77667, 145635, 154737, 309475, 310385, 620771, 621377, 1242737, 1242755, 2485361, 2485475, 4660337, 4970723, 4971025, 9320675, 9903217, 9942051, 19806435, 19864689, 19884107, 39729379

Offset: 1

Ralf Stephan, Jun 18 2025

nonn

Sorted set of all A385109(A198587(i)), i>0.

Cf. A198587, A385109.

N=3;for(n=1, 1000000, s=n; t=0; while(s!=1, if(s%2==0, s=s/2, s=(3*s+1)/2; t++); if(s==1&&t==N&&(n%8==1||n%8==3||n%8==7), print1(n,", ") ); ))

a(22)-a(31) from Michel Marcus, Jun 19 2025

a(32)-a(35) added using A198587 by Jinyuan Wang, Jun 26 2025

A062053 Numbers with exactly 3 odd integers in their Collatz (or 3x+1) trajectory.

Original entry on oeis.org

3, 6, 12, 13, 24, 26, 48, 52, 53, 96, 104, 106, 113, 192, 208, 212, 213, 226, 227, 384, 416, 424, 426, 452, 453, 454, 768, 832, 848, 852, 853, 904, 906, 908, 909, 1536, 1664, 1696, 1704, 1706, 1808, 1812, 1813, 1816, 1818, 3072, 3328, 3392, 3408, 3412, 3413, 3616

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 (A006370).
The Collatz trajectory of n is obtained by applying f repeatedly to n until 1 is reached.
A078719(a(n)) = 3; A006667(a(n)) = 2.

			The Collatz trajectory of 3 is (3,10,5,16,8,4,2,1), which contains 3 odd integers.

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.

Reinhard Zumkeller and David A. Corneth, Table of n, a(n) for n = 1..16191 (first 250 terms from Reinhard Zumkeller, terms < 10^25)
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. A006370, A078719, A006667.
Cf. A000079, A062052, A062054, A062055, A062056, A062057, A062058, A062059, A062060.
Cf. A198584 (this sequence without the even numbers).
See also A198587.
Column k=3 of A354236.

import Data.List (elemIndices)
a062053 n = a062053_list !! (n-1)
a062053_list = map (+ 1) $ elemIndices 3 a078719_list
-- Reinhard Zumkeller, Oct 08 2011

Collatz[n_?OddQ] := (3n + 1)/2; Collatz[n_?EvenQ] := n/2; oddIntCollatzCount[n_] := Length[Select[NestWhileList[Collatz, n, # != 1 &], OddQ]]; Select[Range[4000], oddIntCollatzCount[#] == 3 &] (* Alonso del Arte, Oct 28 2011 *)

The two formulas giving this sequence are listed in Corollary 3.1 and Corollary 3.2 in J. R. Goodwin with the following caveats: the value x cannot equal zero in Corollary 3.2, one must multiply the formulas by all powers of 2 (2^1, 2^2, ...) to get the evens. - Jeffrey R. Goodwin, Oct 26 2011

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

Original entry on oeis.org

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

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A385110 Terms of A198587 congruent {1, 3, 7} (mod 8).

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A062053 Numbers with exactly 3 odd integers in their Collatz (or 3x+1) trajectory.

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

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