A269808 - OEIS

%I #27 Nov 05 2018 21:18:29

%S 55,65,91,110,115,121,130,137,165,182,195,205,213,220,221,230,235,242,

%T 260,273,274,295,330,335,337,345,355,361,363,364,390,391,403,407,410,

%U 411,419,426,440,442,460,467,470,481,484,485,495,497,503,505,517,520,546

%N Numbers having harmonic fractility A270000(n) = 5.

%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="/A269808/b269808.txt">Table of n, a(n) for n = 1..115</a>

%e Nested interval sequences NI(k/m) for m = 55:

%e The 5 equivalence classes are represented by

%e NI(1/55) = (55, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...),

%e NI(2/55) = (27, 2, 1, 1, 1, 1, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, ...),

%e NI(4/55) = (13, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1,...),

%e NI(6/55) = (9, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, ...),

%e NI(22/55) = (2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...).

%e For example, NI(3/55) = (18, 1, 3, 1, 2, 1, 55, 1, 1, 1, ...) is equivalent to NI(1/55).

%o (PARI) select( is_A269808(n)=A270000(n)==5, [1..550]) \\ _M. F. Hasler_, Nov 05 2018

%Y Cf. A269804, A269805, A269806, A269807, A269809 (numbers with harmonic fractility 1, 2, ..., 6), 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 27 2018

%E Edited by _M. F. Hasler_, Nov 05 2018