A269804 - OEIS

%I #30 Nov 05 2018 21:23:52

%S 2,3,4,6,7,8,9,12,14,16,17,18,21,24,27,28,31,32,34,36,42,48,49,51,54,

%T 56,62,63,64,68,72,81,84,93,96,98,102,108,112,113,124,126,128,136,144,

%U 147,151,153,162,168,186,189,192,196,204,216,224,226,241,243

%N Numbers having harmonic fractility 1, cf. A270000.

%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 For harmonic fractility, r(n) = 1/n, n(j+1) = floor(L(j)/(x - Sum_{i=1..j} L(i-1)/(n(i)+1))) for all j >= 0, L(0) = 1. - _M. F. Hasler_, Nov 05 2018

%H Jack W Grahl, <a href="/A269804/b269804.txt">Table of n, a(n) for n = 1..117</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 = 6:

%e NI(1/6) = (6, 1, 1, 1, 1, 1,...)

%e NI(2/6) = (3, 1, 1, 1, 1, 1,...)

%e NI(3/6) = (2, 1, 1, 1, 1, 1,...)

%e NI(4/6) = (1, 3, 1, 1, 1, 1,...)

%e NI(5/6) = (1, 1, 3, 1, 1, 1,...):

%e There is only one equivalence class, so that the fractility of 6 is 1.

%t (* see A270000 *)

%o (PARI) select( is_A269804(n)=A270000(n)==1, [1..250]) \\ _M. F. Hasler_, Nov 05 2018

%Y Cf. A269805, A269806, A269807, A269808, A269809 (numbers with harmonic fractility 2, ..., 6), A270000 (harmonic fractility of n).

%K nonn

%O 1,1

%A _Clark Kimberling_ and _Peter J. C. Moses_, Mar 05 2016

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