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%I #35 Nov 05 2018 21:18:20
%S 77,95,131,145,154,190,203,209,231,247,262,275,285,290,299,308,329,
%T 377,380,393,406,418,431,435,437,443,462,494,524,529,539,545,550,559,
%U 570,580,595,598,609,616,627,658,685,689,693,705,737,741,754,760,767,786
%N Numbers having harmonic fractility A270000(n) = 6.
%C 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.
%C 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.
%C 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
%H Jack W Grahl, <a href="/A269809/b269809.txt">Table of n, a(n) for n = 1..73</a>
%H Peter J. C. Moses, Clark Kimberling, <a href="http://math.colgate.edu/~integers/r46/r46.Abstract.html">Nested interval sequences of positive real numbers</a>, Integers 17 (2017), #A46.
%e Nested interval sequences NI(k/m) for m = 77:
%e The 6 equivalence classes are represented by
%e NI(1/77) = (77, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...),
%e 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),
%e NI(3/77) = (25, 3, 1, 1, 1, 5, 15, 1, 5, 15, 1, 5, 15, 1, 5, 15, 1, 5, 15, ...),
%e NI(8/77) = (9, 2, 9, 2, 9, 2, 9, 2, 9, 2, 9, 2, 9, 2, 9, 2, 9, 2, 9, 2, 9, ...),
%e NI(10/77) = (7, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, ...),
%e NI(14/77) = (5, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, ...).
%e 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
%o (PARI) select( is_A269809(n)=A270000(n)==6, [1..800]) \\ _M. F. Hasler_, Nov 05 2018
%Y Cf. A269804, A269805, A269806, A269807, A269808 (numbers with harmonic fractility 1, 2, 3, 4, 5, respectively); A270000 (harmonic fractility of n).
%K nonn
%O 1,1
%A _Clark Kimberling_ and _Peter J. C. Moses_, Mar 05 2016
%E More terms from _Jack W Grahl_, Jun 28 2018
%E Edited by _M. F. Hasler_, Nov 05 2018
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