

A022328


Exponent of 2 (value of i) in nth number of form 2^i*3^j (see A003586).


21



0, 1, 0, 2, 1, 3, 0, 2, 4, 1, 3, 0, 5, 2, 4, 1, 6, 3, 0, 5, 2, 7, 4, 1, 6, 3, 0, 8, 5, 2, 7, 4, 1, 9, 6, 3, 0, 8, 5, 2, 10, 7, 4, 1, 9, 6, 3, 11, 0, 8, 5, 2, 10, 7, 4, 12, 1, 9, 6, 3, 11, 0, 8, 5, 13, 2, 10, 7, 4, 12, 1, 9, 6, 14, 3, 11, 0, 8, 5, 13, 2, 10, 7, 15, 4, 12, 1, 9, 6, 14, 3, 11, 0, 8, 16, 5, 13, 2
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OFFSET

1,4


COMMENTS

This is the signature sequence of log(3)/log(2) and is a fractal sequence; e.g., if the first occurrence of each n is removed, the resulting sequence is the original sequence.
Moreover, if the sequence is partitioned into segments starting with 0 as follows:
0,1
0,2,1,3
0,2,4,1,3
0,5,2,4,1,6,3,
and so on, then deleting the greatest number in each segment leaves
0
0,2,1
0,2,1,3
0,5,2,4,1,3,
and so on, which, concatenated to (0,0,2,1,0,2,1,3,0,5,2,4,1,3,...), is another fractal sequence, in today's usual meaning of that term. When introduced in 1995, one of the defining properties of a fractal sequence was, essentially, that before each n appears, every k < n must have already appeared; this requirement ensures that the sequence yields a dispersion; e.g., A114577 yields A114537. However, the usual meaning of "fractal sequence" nowadays is simply "a sequence that contains itself as a proper subsequence". It is proposed here that the original version be renamed "strongly fractal". Thus, the operations called upper trimming and lower trimming (e.g., A084531, A167237), when applied to strongly fractal sequences, yield strongly fractal sequences. The operation introduced here, which can be called "segmentupper trimming", carries fractal sequences to fractal sequences, but not strongly fractal to strongly fractal.
Associated with the signature sequence S of each positive irrational number is an interspersion (or equivalently, a dispersion), in which row n >= 0 consists of the positions of n in S. The interspersion associated with the signature sequence of log(3)/log(2) is A255975.
(End)


LINKS



FORMULA



MATHEMATICA

t = Sort[Flatten[Table[2^i 3^j, {i, 0, 200}, {j, 0, 200}]]];
Table[IntegerExponent[t[[n]], 2], {n, 1, 200}] (* A022338 *)


PROG

(Haskell)
import Data.Set (singleton, deleteFindMin, insert)
a022328 n = a022328_list !! (n1)
(a022328_list, a022329_list) = unzip $ f $ singleton (1, (0, 0)) where
f s = (i, j) :
f (insert (2 * y, (i + 1, j)) $ insert (3 * y, (i, j + 1)) s')
where ((y, (i, j)), s') = deleteFindMin s


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



