A269806 Numbers having harmonic fractility A270000(n) = 3.

11, 13, 19, 22, 23, 25, 26, 29, 33, 35, 38, 39, 44, 46, 47, 50, 52, 53, 57, 58, 66, 67, 69, 70, 75, 76, 78, 79, 83, 87, 88, 89, 92, 94, 99, 100, 104, 105, 106, 114, 116, 117, 119, 125, 132, 133, 134, 138, 140, 149, 150, 152, 155, 156, 158, 159, 161, 166, 171

Offset: 1

Author

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

Keywords

• nonn

Comments

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 identical. 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 = 11:
NI(1/11) = (11,1, 1, 1, 1, 1, 1, 1, ...),
NI(2/11) = (5, 2, 1, 2, 1, 2, 1, 1, 2, ...),
NI(3/11) = (3, 3, 3, 3, 3, 3, 3, 3, 3, ...),
NI(4/11) = (2, 5, 2, 1, 2, 1, 2, 1, 2, ...),
NI(5/11) = (2, 1, 2, 1, 2, 1, 2, 1, 2, ...) equivalent to NI(4/11),
NI(6/11) = (1, 11, 1, 1, 1, 1, 1, 1, ...) equivalent to NI(1/11),
NI(7/11) = (1, 3, 3, 3, 3, 3, 3, 3, 3, ...) equivalent to NI(3/11),
NI(8/11) = (1, 2, 1, 2, 1, 2, 1, 2, 1, ...) equivalent to NI(4/11),
NI(9/11) = (1, 1, 3, 3, 3, 3, 3, 3, 3, ...) equivalent to NI(3/11),
NI(10/11) = (1, 1, 1, 3, 3, 3, 3, 3, ...) equivalent to NI(3/11).
So there are 3 equivalence classes for m = 11, and the fractility of 11 is 3.
		

Links

• Jack W Grahl, Table of n, a(n) for n = 1..299

Crossrefs

Cf. A269804, A269805, A269807, A269808, A269809 (numbers with harmonic fractility 1, 2, 4, 5, 6, respectively); A270000 (harmonic fractility of n).

Programs

PARI

select( is_A269806(n)=A270000(n)==3, [1..300]) \\ M. F. Hasler, Nov 05 2018

Extensions

Edited by M. F. Hasler, Nov 05 2018