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
The indices of the powers of 3 in A006899. - Ruud H.G. van Tol, Nov 02 2022
LINKS
T. D. Noe, Table of n, a(n) for n = 1..1000
FORMULA
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 *)
PROG
(PARI) a(n)=logint(3^(n-1), 2)+n \\ Ruud H.G. van Tol, Nov 04 2022
CROSSREFS
KEYWORD
nice,nonn
AUTHOR
T. D. Noe, Sep 06 2006
EXTENSIONS
Comment corrected and edited by Jon E. Schoenfield, Feb 27 2014
STATUS
approved