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A122437 Allowable values of the "dropping time" of the Collatz (3x+1) iteration. 9
1, 3, 6, 8, 11, 13, 16, 19, 21, 24, 26, 29, 32, 34, 37, 39, 42, 44, 47, 50, 52, 55, 57, 60, 63, 65, 68, 70, 73, 75, 78, 81, 83, 86, 88, 91, 94, 96, 99, 101, 104, 106, 109, 112, 114, 117, 119, 122, 125, 127, 130, 132, 135, 138, 140, 143, 145, 148, 150, 153, 156, 158, 161 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Only these numbers appear in A060445, which tabulates the "dropping time" of odd numbers. Note that all even numbers have a "dropping time" of 1.

a(n) is also the number of binary digits of 6^(n-1); for example, a(4)=8 since 6^(4-1)=216 in binary is 11011000, an 8-digit number. - Julio Cesar de la Yncera, Mar 28 2009

A positive integer (x) is an allowable value if and only if (x-1)/(1+log(2)/log(3))-floor(x/(1+log(2)/log(3))) is not negative. - K. Spage, Oct 22 2009

Here the word "allowable" means that it is necessary for a sequence of iterates starting from odd value m to arrive at a value x = f^{floor(1+n+n*log(3)/log(2))}(m) < m, where n gives the number of odds in such a sequence including m, to have undergone precisely floor(1+n+n*log(3)/log(2)) iterations of f, where f(2*m)=m, f(2*m+1)=6*m+4. However, the formula for a(n+1) does not fully account for the order of odds and evens in such a sequence because it does not account for the effects of the "+1". Thus it is unknown whether it maximizes the value x for all values m. For example, fix m = 1 and the "+1" is enough to give the trivial cycle. So it is possible that for some m we have f^{floor(1+n+n*log(3)/log(2))}(m) >= m. - Jeffrey R. Goodwin, Aug 24 2011

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

FORMULA

a(1) = 1, a(n+1) = a(n)+A022921(n-1)+1.

a(n+1) = floor(1+n+n*log(3)/log(2)). - T. D. Noe, Sep 08 2006

a(n) = floor((1+log(2)/log(3))*A020914(n-1)). - K. Spage, Oct 22 2009

a(n) = A020914(n+1)+n-1. - K. Spage, Oct 23 2009

MATHEMATICA

Floor[1+Range[0, 100]*(1+Log[2, 3])] (* T. D. Noe, Sep 08 2006 *)

Map[Length[RealDigits[ #, 2][[1]]] &, Table[10^i, {i, 0, 50}]] (* Julio Cesar de la Yncera, Mar 28 2009 *)

CROSSREFS

Cf. A022921 (number of 2^m between 3^n and 3^(n+1)), A122442 (least k having dropping time a(n)).

Cf.

Sequence in context: A139477 A210570 A224839 * A286988 A090848 A184657

Adjacent sequences:  A122434 A122435 A122436 * A122438 A122439 A122440

KEYWORD

nice,nonn

AUTHOR

T. D. Noe, Sep 06 2006

EXTENSIONS

Comment corrected and edited by Jon E. Schoenfield, Feb 27 2014

STATUS

approved

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Last modified May 28 16:22 EDT 2017. Contains 287241 sequences.