login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A269808 Numbers having harmonic fractility A270000(n) = 5. 7
55, 65, 91, 110, 115, 121, 130, 137, 165, 182, 195, 205, 213, 220, 221, 230, 235, 242, 260, 273, 274, 295, 330, 335, 337, 345, 355, 361, 363, 364, 390, 391, 403, 407, 410, 411, 419, 426, 440, 442, 460, 467, 470, 481, 484, 485, 495, 497, 503, 505, 517, 520, 546 (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.

In the case of harmonic fractility, r(n) = 1/n, we have n(j+1) = floor(L(j)/(x -Sum_{i=1..j} L(i-1)/(n(i)+1))) for j >= 0, L(0) = 1. - M. F. Hasler, Nov 05 2018

LINKS

Jack W Grahl, Table of n, a(n) for n = 1..115

EXAMPLE

Nested interval sequences NI(k/m) for m = 55:

The 5 equivalence classes are represented by

NI(1/55) = (55, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...),

NI(2/55) = (27, 2, 1, 1, 1, 1, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, ...),

NI(4/55) = (13, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1,...),

NI(6/55) = (9, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, ...),

NI(22/55) = (2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...).

For example, NI(3/55) = (18, 1, 3, 1, 2, 1, 55, 1, 1, 1, ...) is equivalent to NI(1/55).

PROG

(PARI) select( is_A269808(n)=A270000(n)==5, [1..550]) \\ M. F. Hasler, Nov 05 2018

CROSSREFS

Cf. A269804, A269805, A269806, A269807, A269809 (numbers with harmonic fractility 1, 2, ..., 6), A270000 (harmonic fractility of n).

Sequence in context: A043179 A043959 A285804 * A004434 A168109 A116055

Adjacent sequences:  A269805 A269806 A269807 * A269809 A269810 A269811

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

More terms from Jack W Grahl, Jun 27 2018

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 December 2 08:31 EST 2021. Contains 349437 sequences. (Running on oeis4.)