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A060445
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"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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11
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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
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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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
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EXAMPLE
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3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2, taking 6 steps, so a(1) = 6.
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MATHEMATICA
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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 *)
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PROG
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(Haskell)
a060445 0 = 0
a060445 n = length $ takeWhile (>= n') $ a070165_row n'
where n' = 2 * n + 1
-- Reinhard Zumkeller, Mar 11 2013
(Python)
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 n<N: break
return x
print [a(n) for n in xrange(0, 101)] # Indranil Ghosh, Apr 22 2017
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CROSSREFS
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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.
Sequence in context: A272549 A112456 A060534 * A131894 A040033 A165998
Adjacent sequences: A060442 A060443 A060444 * A060446 A060447 A060448
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane, Apr 07 2001
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EXTENSIONS
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More terms from Jason Earls (zevi_35711(AT)yahoo.com), Apr 08 2001 and from Michel ten Voorde (seqfan(AT)tenvoorde.org) Apr 09 2001. Still more terms from Larry Reeves (larryr(AT)acm.org), Apr 12 2001
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STATUS
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approved
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