A269989 Odds fractility of n.

1, 2, 2, 3, 3, 4, 3, 3, 6, 6, 5, 6, 7, 6, 8, 7, 5, 8, 8, 8, 10, 10, 10, 10, 11, 5, 13, 11, 9, 15, 13, 11, 14, 15, 10, 16, 15, 11, 15, 18, 14, 18, 18, 10, 23, 17, 14, 18, 15, 16, 25, 20, 10, 20, 24, 15, 25, 23, 16, 27, 27, 14, 23, 24, 21, 26, 27, 25, 26, 29

Offset

2,2

Comments

In order to define (odds) fractility of an integer n > 1, we first define nested interval sequences. Suppose that r = (r(n)) is a sequence satisfying (i) 1 = r(1) > r(2) > r(3) > ... and (ii) r(n) -> 0. For x in (0,1], let n(1) be the index n such that r(n+1) , x <= r(n), and let L(1) = r(n(1))-r(n(1)+1). Let n(2) be the index n such that r(n(1)+1) < x <= r(n(1)+1) + L(1)r(n), and let L(2) = (r(n(2))-r(r(n)+1))L(1). Continue inductively to obtain the sequence (n(1), n(2), n(3), ... ), the r-nested interval sequence of x.

For fixed r, call x and y equivalent if NI(x) and NI(y) are eventually identical. For n > 1, the r-fractility of n is the number of equivalence classes of sequences NI(m/n) for 0 < m < n. Taking r = (1/1, 1/3, 1/5, 1/7, 1/9, ... ) gives odds fractilily.

binary fractility: A269570

factorial fractility: A269982

harmonic fractility: A270000

primes fractility: A269990

Links

Table of n, a(n) for n=2..71.

Example

NI(1/7) = (4,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...)
NI(2/7) = (2,1,1,3,1,1,1,1,1,1,2,1,1,2,2,1,2,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,2,1,2,1,1,1,19,1,30,1,2,2,1,10,1,1,3,1,...)
NI(3/7) = (1,4,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...)
NI(4/7) = (1,1,14,1,2,3,1,2,1,1,1,1,1,2,1,3,2,1,6,1,1,11,1,1,1,1,1,1,1,12,2,1,1,1,2,3,1,1,1,1,1,6,1,1,1,1,2,3,1,7,...)
NI(5/7) = (1,1,1,14,1,2,3,1,2,1,1,1,1,1,2,1,3,2,1,6,1,1,11,1,1,1,1,1,1,1,12,2,1,1,1,2,3,1,1,1,1,1,6,1,1,1,1,2,3,1,...)
NI(6/7) = (1,1,1,1,2,1,1,11,1,2,1,1,1,1,1,1,1,6,1,7,1,1,1,1,1,1,1,2,1,1,6,1,1,1,194,1,2,7,6,2,1,1,1,1,1,1,3,1,2,1,...);
there are 4 equivalence classes:  {1/7,3/7},{2/7},{4,5},{6},so that a(7) = 4.

Crossrefs

Cf. A005408, A269570, A269982, A270000, A269990.

Sequence in context: A324861 A324863 A332894 * A057935 A292042 A282715

Adjacent sequences: A269986 A269987 A269988 * A269990 A269991 A269992

Keyword

nonn,easy

Author

Clark Kimberling and Peter J. C. Moses, Mar 12 2016

Status

approved