A269809 Numbers having harmonic fractility A270000(n) = 6.

77, 95, 131, 145, 154, 190, 203, 209, 231, 247, 262, 275, 285, 290, 299, 308, 329, 377, 380, 393, 406, 418, 431, 435, 437, 443, 462, 494, 524, 529, 539, 545, 550, 559, 570, 580, 595, 598, 609, 616, 627, 658, 685, 689, 693, 705, 737, 741, 754, 760, 767, 786

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 = 77:
The 6 equivalence classes are represented by
NI(1/77) = (77, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...),
NI(2/77) = (38, 2, 1, 1, 1, 1, 2, 3, 8, 2, 2, 1, 1, 1, 1, 2, 3, 8, 2, 2, 1, ...) (period length 9),
NI(3/77) = (25, 3, 1, 1, 1, 5, 15, 1, 5, 15, 1, 5, 15, 1, 5, 15, 1, 5, 15, ...),
NI(8/77) = (9, 2, 9, 2, 9, 2, 9, 2, 9, 2, 9, 2, 9, 2, 9, 2, 9, 2, 9, 2, 9, ...),
NI(10/77) = (7, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, ...),
NI(14/77) = (5, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, ...).
For example, N(4/77) = (19, 1, 2, 1, 1, 1, 15, 1, 5, 15, 1, 5, ...) is equivalent to NI(3/77), and NI(6/77) = (12, 6, 1, 11, 1, 1, 1, ...) is equivalent to NI(1/77). - _M. F. Hasler_, Nov 05 2018
		

Links

• Jack W Grahl, Table of n, a(n) for n = 1..73
• Peter J. C. Moses, Clark Kimberling, Nested interval sequences of positive real numbers, Integers 17 (2017), #A46.

Crossrefs

Cf. A269804, A269805, A269806, A269807, A269808 (numbers with harmonic fractility 1, 2, 3, 4, 5, respectively); A270000 (harmonic fractility of n).

Programs

PARI

select( is_A269809(n)=A270000(n)==6, [1..800]) \\ M. F. Hasler, Nov 05 2018

Extensions

More terms from Jack W Grahl, Jun 28 2018

Edited by M. F. Hasler, Nov 05 2018