This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A269806 #22 Nov 05 2018 21:22:31
%S A269806 11,13,19,22,23,25,26,29,33,35,38,39,44,46,47,50,52,53,57,58,66,67,69,
%T A269806 70,75,76,78,79,83,87,88,89,92,94,99,100,104,105,106,114,116,117,119,
%U A269806 125,132,133,134,138,140,149,150,152,155,156,158,159,161,166,171
%N A269806 Numbers having harmonic fractility A270000(n) = 3.
%C A269806 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 A269806 For fixed r, call x and y equivalent if NI(x) and NI(y) are eventually identical. 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 A269806 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 A269806 Jack W Grahl, <a href="/A269806/b269806.txt">Table of n, a(n) for n = 1..299</a>
%e A269806 Nested interval sequences NI(k/m) for m = 11:
%e A269806 NI(1/11) = (11,1, 1, 1, 1, 1, 1, 1, ...),
%e A269806 NI(2/11) = (5, 2, 1, 2, 1, 2, 1, 1, 2, ...),
%e A269806 NI(3/11) = (3, 3, 3, 3, 3, 3, 3, 3, 3, ...),
%e A269806 NI(4/11) = (2, 5, 2, 1, 2, 1, 2, 1, 2, ...),
%e A269806 NI(5/11) = (2, 1, 2, 1, 2, 1, 2, 1, 2, ...) equivalent to NI(4/11),
%e A269806 NI(6/11) = (1, 11, 1, 1, 1, 1, 1, 1, ...) equivalent to NI(1/11),
%e A269806 NI(7/11) = (1, 3, 3, 3, 3, 3, 3, 3, 3, ...) equivalent to NI(3/11),
%e A269806 NI(8/11) = (1, 2, 1, 2, 1, 2, 1, 2, 1, ...) equivalent to NI(4/11),
%e A269806 NI(9/11) = (1, 1, 3, 3, 3, 3, 3, 3, 3, ...) equivalent to NI(3/11),
%e A269806 NI(10/11) = (1, 1, 1, 3, 3, 3, 3, 3, ...) equivalent to NI(3/11).
%e A269806 So there are 3 equivalence classes for m = 11, and the fractility of 11 is 3.
%o A269806 (PARI) select( is_A269806(n)=A270000(n)==3, [1..300]) \\ _M. F. Hasler_, Nov 05 2018
%Y A269806 Cf. A269804, A269805, A269807, A269808, A269809 (numbers with harmonic fractility 1, 2, 4, 5, 6, respectively); A270000 (harmonic fractility of n).
%K A269806 nonn
%O A269806 1,1
%A A269806 _Clark Kimberling_ and _Peter J. C. Moses_, Mar 05 2016
%E A269806 Edited by _M. F. Hasler_, Nov 05 2018