cp's OEIS Frontend

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

%I A269807 #34 Nov 05 2018 21:20:48
%S A269807 41,71,82,109,123,141,142,157,163,164,169,175,179,181,187,191,197,211,
%T A269807 218,229,246,251,257,265,271,282,284,293,305,311,314,323,326,327,328,
%U A269807 338,341,350,358,362,369,371,374,382,394,395,415,422,423,433,436,445,449
%N A269807 Numbers having harmonic fractility A270000(n) = 4.
%C A269807 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 A269807 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 A269807 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 A269807 Jack W Grahl, <a href="/A269807/b269807.txt">Table of n, a(n) for n = 1..138</a>
%e A269807 Nested interval sequences NI(k/m) for m = 41:
%e A269807 NI(1/41) = (41, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...),
%e A269807 NI(2/41) = (20, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, ...),
%e A269807 NI(3/41) = (13, 3, 1, 1, 4, 2, 2, 20, 1, 1, 1, 2, 2, 1, 1, 1, 2, ...) ~ NI(2/41),
%e A269807 NI(4/41) = (10, 1, 2, 1, 1, 8, 1, 1, 8, 1, 1, 8, 1, 1, 8, 1, 1, 8, 1, 1, ...),
%e A269807 NI(5/41) = (6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, ...).
%e A269807 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).
%e A269807 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.
%o A269807 (PARI) select(is_A269807(n)=A270000(n)==4, [1..450]) \\ _M. F. Hasler_, Nov 05 2018
%Y A269807 Cf. A269804, A269805, A269806, A269808, A269809 (numbers with harmonic fractility 1, 2, ..., 6), A270000 (harmonic fractility of n).
%K A269807 nonn
%O A269807 1,1
%A A269807 _Clark Kimberling_ and _Peter J. C. Moses_, Mar 05 2016
%E A269807 More terms from _Jack W Grahl_, Jun 27 2018
%E A269807 Edited by _M. F. Hasler_, Nov 05 2018