A269807 Numbers having harmonic fractility A270000(n) = 4.

41, 71, 82, 109, 123, 141, 142, 157, 163, 164, 169, 175, 179, 181, 187, 191, 197, 211, 218, 229, 246, 251, 257, 265, 271, 282, 284, 293, 305, 311, 314, 323, 326, 327, 328, 338, 341, 350, 358, 362, 369, 371, 374, 382, 394, 395, 415, 422, 423, 433, 436, 445, 449

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 = 41:
NI(1/41) = (41, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...),
NI(2/41) = (20, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, ...),
NI(3/41) = (13, 3, 1, 1, 4, 2, 2, 20, 1, 1, 1, 2, 2, 1, 1, 1, 2, ...) ~ NI(2/41),
NI(4/41) = (10, 1, 2, 1, 1, 8, 1, 1, 8, 1, 1, 8, 1, 1, 8, 1, 1, 8, 1, 1, ...),
NI(5/41) = (6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, ...).
Any further NI(k/41) is equivalent to one of the  above, e.g., NI(40/11) = (1, 1, 1, 1, 1, 4, 2, 2, 20, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, ...) ~ NI(2/41).
Thus, the number of equivalence classes is 4 (represented by 1/41, 2/41, 4/41 and 5/41), so that the fractility of 41 is 4.
		

Links

• Jack W Grahl, Table of n, a(n) for n = 1..138

Crossrefs

Cf. A269804, A269805, A269806, A269808, A269809 (numbers with harmonic fractility 1, 2, ..., 6), A270000 (harmonic fractility of n).

Programs

PARI

select(is_A269807(n)=A270000(n)==4, [1..450]) \\ M. F. Hasler, Nov 05 2018

Extensions

More terms from Jack W Grahl, Jun 27 2018

Edited by M. F. Hasler, Nov 05 2018