login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A269804 Numbers having harmonic fractility 1, cf. A270000. 7
2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 17, 18, 21, 24, 27, 28, 31, 32, 34, 36, 42, 48, 49, 51, 54, 56, 62, 63, 64, 68, 72, 81, 84, 93, 96, 98, 102, 108, 112, 113, 124, 126, 128, 136, 144, 147, 151, 153, 162, 168, 186, 189, 192, 196, 204, 216, 224, 226, 241, 243 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

In order to define (harmonic) fractility of an integer m > 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 largest index n such that x <= r(n(1)+1) + L(1)*r(n), and let L(2) = (r(n(2))-r(n(2)+1))*L(1).  Continue inductively to obtain the sequence (n(1), n(2), n(3), ... ) =: NI(x), the r-nested interval sequence of x.

For fixed r, call x and y equivalent if NI(x) and NI(y) are eventually equal (up to an offset).  For m > 1, the r-fractility of m is the number of equivalence classes of sequences NI(k/m) for 0 < k < m.  Taking r = (1/1, 1/2, 1/3, 1/4, ...) gives harmonic fractility.

For harmonic fractility, r(n) = 1/n, n(j+1) = floor(L(j)/(x - Sum_{i=1..j} L(i-1)/(n(i)+1))) for all j >= 0, L(0) = 1. - M. F. Hasler, Nov 05 2018

LINKS

Jack W Grahl, Table of n, a(n) for n = 1..117

Peter J. C. Moses, Clark Kimberling, Nested interval sequences of positive real numbers, Integers 17 (2017), #A46.

EXAMPLE

Nested-interval sequences NI(k/m) for m = 6:

NI(1/6) = (6, 1, 1, 1, 1, 1,...)

NI(2/6) = (3, 1, 1, 1, 1, 1,...)

NI(3/6) = (2, 1, 1, 1, 1, 1,...)

NI(4/6) = (1, 3, 1, 1, 1, 1,...)

NI(5/6) = (1, 1, 3, 1, 1, 1,...):

There is only one equivalence class, so that the fractility of 6 is 1.

MATHEMATICA

(* see A270000 *)

PROG

(PARI) select( is_A269804(n)=A270000(n)==1, [1..250]) \\ M. F. Hasler, Nov 05 2018

CROSSREFS

Cf. A269805, A269806, A269807, A269808, A269809 (numbers with harmonic fractility 2, ..., 6), A270000 (harmonic fractility of n).

Sequence in context: A038032 A010408 A010455 * A318400 A219174 A108319

Adjacent sequences:  A269801 A269802 A269803 * A269805 A269806 A269807

KEYWORD

nonn

AUTHOR

Clark Kimberling and Peter J. C. Moses, Mar 05 2016

EXTENSIONS

Edited by M. F. Hasler, Nov 05 2018

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 25 14:43 EDT 2019. Contains 324352 sequences. (Running on oeis4.)