A269808 Numbers having harmonic fractility A270000(n) = 5.

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

Offset: 1

Author

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

Keywords

• nonn

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

Examples

			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).
		

Links

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

Crossrefs

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

Programs

PARI

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

Extensions

More terms from Jack W Grahl, Jun 27 2018

Edited by M. F. Hasler, Nov 05 2018