A060565 -id:A060565 - cp's OEIS frontend

A060445 "Dropping time" in 3x+1 problem starting at 2n+1 (number of steps to reach a lower number than starting value). Also called glide(2n+1).

0, 6, 3, 11, 3, 8, 3, 11, 3, 6, 3, 8, 3, 96, 3, 91, 3, 6, 3, 13, 3, 8, 3, 88, 3, 6, 3, 8, 3, 11, 3, 88, 3, 6, 3, 83, 3, 8, 3, 13, 3, 6, 3, 8, 3, 73, 3, 13, 3, 6, 3, 68, 3, 8, 3, 50, 3, 6, 3, 8, 3, 13, 3, 24, 3, 6, 3, 11, 3, 8, 3, 11, 3, 6, 3, 8, 3, 65, 3, 34, 3, 6, 3, 47, 3, 8, 3, 13, 3, 6, 3, 8, 3

Offset: 0

N. J. A. Sloane, Apr 07 2001

If the starting value is even then of course the next step in the trajectory is smaller (cf. A102419).

The dropping time can be made arbitrarily large: If the starting value is of form n(2^m)-1 and m > 1, the next value is 3n(2^m)-3+1. That divided by 2 is 3n(2^(m-1))-1. It is bigger than the starting value and of the same form - substitute 3n -> n and m-1 -> m, so recursively get an increasing subsequence of m odd values. The dropping time is obviously longer than that. This holds even if Collatz conjecture were refuted. For example, m=5, n=3 -> 95, 286, 143, 430, 215, 646, 323, 970, 485, 1456, 728, 364, 182, 91. So the subsequence in reduced Collatz variant is 95, 143, 215, 323, 485. - Juhani Heino, Jul 21 2017

			3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2, taking 6 steps, so a(1) = 6.

T. D. Noe, Table of n, a(n) for n = 0..10000
Jason Holt, Plot of first 1 billion terms, log scale on x axis
Jason Holt, Plot of first 10 billion terms, log scale on x axis
Eric Roosendaal, On the 3x + 1 problem
Index entries for sequences related to 3x+1 (or Collatz) problem

A060565 gives the first lower number that is reached. Cf. A060412-A060415, A217934.
See A074473, A102419 for other versions of this sequence.
Cf. A122437 (allowable dropping times), A122442 (least k having dropping time A122437(n)).
Cf. A070165.

a060445 0 = 0
a060445 n = length $ takeWhile (>= n') $ a070165_row n'
            where n' = 2 * n + 1
-- Reinhard Zumkeller, Mar 11 2013

nxt[n_]:=If[OddQ[n],3n+1,n/2]; Join[{0},Table[Length[NestWhileList[nxt, n,#>=n&]]-1, {n,3,191,2}]]  (* Harvey P. Dale, Apr 23 2011 *)

def a(n):
    if n<1: return 0
    n=2*n + 1
    N=n
    x=0
    while True:
        if n%2==0: n//=2
        else: n = 3*n + 1
        x+=1
        if nIndranil Ghosh, Apr 22 2017

More terms from Jason Earls, Apr 08 2001 and from Michel ten Voorde Apr 09 2001

Still more terms from Larry Reeves (larryr(AT)acm.org), Apr 12 2001

A171490 Numbers for which the smallest number of steps to reach 1 in "3x+1" (or Collatz) problem is a prime.

Original entry on oeis.org

1, 5, 7, 12, 14, 16, 29, 51, 56, 58, 60, 64, 65, 67, 74, 75, 78, 83, 87, 90, 100, 102, 104, 106, 109, 115, 118, 119, 122, 128, 130, 132, 134, 141, 142, 147, 161, 166, 173, 176, 187, 188, 200, 212, 219, 221, 231, 234, 239, 241, 251, 259, 264, 293, 313, 314, 316

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Dec 10 2009

Positions of primes in A033491. [R. J. Mathar, Nov 01 2010]

			1st Collatz sequence with a(1)=1 step starts with 2=prime(1): 2-1;
1st Collatz sequence with a(3)=7 steps starts with 3=prime(2): 3-10-5-16-8-4-2-1;
prime(6)=13 has Collatz sequence with 9 steps: 13-40-20-10-5-16-8-4-2-1, so has the smaller composite 12 < 13: 12-6-3-10-5-16-8-4-2-1 => 9 not a term of sequence;
1st Collatz sequence with a(5)=14 steps starts with 11=prime(5): 11-34-17-52-26-13-40-20-10-5-16-8-4-2-1.

R. K. Guy, "Collatz's Sequence" in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 215-218, 1994
Clifford A. Pickover, Wonders of Numbers, Oxford University Press, pp. 116-118, 2001

J. C. Lagarias, The 3x+1 Problem and its Generalizations, Amer. Math. Monthly 92, 3-23, 1985

Cf. A070905, A033491, A088975, A126727, A060565

Terms > 187 from R. J. Mathar, Nov 01 2010

Name edited by Michel Marcus, Jul 07 2018

A350082 Smallest odd number > n with n in its Collatz successors, or 0 if no such odd number exists. a(1) = 1.

Original entry on oeis.org

1, 3, 0, 5, 7, 0, 9, 9, 0, 11, 19, 0, 17, 37, 0, 17, 19, 0, 25, 23, 0, 25, 27, 0, 33, 29, 0, 37, 33, 0, 41, 75, 0, 37, 41, 0, 43, 39, 0, 41, 109, 0, 57, 51, 0, 47, 55, 0, 57, 133, 0, 57, 55, 0, 73, 57, 0, 59, 123, 0, 63, 109, 0, 75, 115, 0, 89, 181, 0, 71, 73

Offset: 1

Ruud H.G. van Tol, Jan 22 2022

nonn

a(n) = 0 when n == 0 (mod 3) since such an n has no odd predecessor (only n*2^x). But n !== 0 (mod 3) always has some odd predecessor > n.

			a(2) = 3, because 3 is the smallest odd number > 2 that has 2 as a successor: 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2.
a(3) = 0 because 3 is not a successor of anything. A060565 contains no 3, nor multiples of 3.
a(11) = 19, because the trajectories of 13, 15, 17 don't contain 11, and 11 is a successor of 19:
  13 -> 40..5 -> 16..1;
  15 -> 46..23 -> 70..35 -> 106..53 -> 160..5 -> 16..1;
  17 -> 52..13;
  19 -> 58..29 -> 88..11.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A060565.

a(n)= if(1==n, return(1)); if(!(n%3), return(0)); my(v0=if(n%2, n+2, n+1)); while(1, my(v=v0); while(v>1 && v!=n, v=if(v%2, 3*v+1, v/2)); if(v==n, return(v0)); v0+=2)

A171619 Primes in A171490.

Original entry on oeis.org

5, 7, 29, 67, 83, 109, 173, 239, 241, 251, 293, 313, 337, 367, 571, 613, 769, 821, 877, 941, 947, 1031, 1069, 1103, 1511, 1693, 1759, 1901, 2011

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Dec 13 2009

Terms of sequence are primes in growing order where smallest number of steps m to reach 1 in "3x+1" (or Collatz) problem is a prime too.

			(1) 1st Collatz sequence with 5=prime(3) steps starts with 5=prime(3): 5-16-8-4-2-1, gives a(1)=5.
(2) 1st Collatz sequence with 7=prime(4) steps starts with 3=prime(2): 3-10-5-16-8-4-2-1, gives a(2)=7.
(3) 1st Collatz sequence with 29=prime(10) steps starts with 43=prime(14): 43-130-65-196-98-49-148-74-37-112-56-28-14-7-22-11-34-17-52-26-13-40-20-10-5-16-8-4-2-1, gives a(3)=29.
(4) List of prime steps m for above a(n): 5, 3, 43, 167, 233, 41, 937, 14831, 9887, 7963, 73063, 45127, 78791, 225023, 6956969, 10998599, 126357223, 859130059, 2845683047, 322623647, 95592191, 8363817307, 28677246203, 38590505339, 35521451596571, 478672174364191, 1168778549494463, 6376392739978081, 103147916159472367.

R. K. Guy, "Collatz's Sequence" in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 215-218, 1994.
Clifford A. Pickover, Wonders of Numbers, Oxford University Press, pp. 116-118, 2001.
Guenther J. Wirsching, The Dynamical System Generated by the 3n+1 Function, Springer-Verlag, Berlin, 1998.

Eric Roosendaal, 3x+1 Class Records
Eric Weisstein's World of Mathematics, Collatz Problem

Cf. A070905, A033491, A088975, A126727, A060565, A171490.

Missing term a(7)=173 inserted by Georg Fischer, Oct 26 2022

a(23)-a(29) (using Eric Roosendaal's data) by Tyler Busby, Feb 11 2023

cp's OEIS Frontend

A060445 "Dropping time" in 3x+1 problem starting at 2n+1 (number of steps to reach a lower number than starting value). Also called glide(2n+1).

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A171490 Numbers for which the smallest number of steps to reach 1 in "3x+1" (or Collatz) problem is a prime.

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A350082 Smallest odd number > n with n in its Collatz successors, or 0 if no such odd number exists. a(1) = 1.

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A171619 Primes in A171490.

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