A269805 Numbers having harmonic fractility A270000(n) = 2.

5, 10, 15, 20, 30, 37, 40, 43, 45, 59, 60, 61, 73, 74, 80, 85, 86, 90, 97, 101, 103, 107, 111, 118, 120, 122, 127, 129, 135, 139, 146, 148, 160, 167, 170, 172, 177, 180, 183, 194, 199, 202, 206, 214, 219, 222, 236, 240, 244, 254, 255, 258, 270, 277, 278, 291

Offset

1,1

Comments

In order to define (harmonic) fractility of an integer n > 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.

For fixed r, call x and y equivalent if NI(x) and NI(y) are eventually equal (up to an offset). For n > 1, the r-fractility of n is the number of equivalence classes of sequences NI(m/n) for 0 < m < n. Taking r = (1/1, 1/2, 1/3, 1/4, ...) gives harmonic fractility.

Links

Jack W Grahl, Table of n, a(n) for n = 1..189

Example

Nested interval sequences NI(k/m) for m = 5:
NI(1/5) = (5, 1, 1, 1, 1, 1,...),
NI(2/5) = (2, 2, 2, 2, 2, 2,...),
NI(3/5) = (1, 5, 1, 1, 1, 1,...),
NI(4/5) = (1, 1, 5, 1, 1, 1,...),
so that there are 2 equivalence classes for n = 5, and the fractility of 5 is 2.

Prog

(PARI) select( is_A269805(m)=A270000(n)==2, [1..300]) \\ M. F. Hasler, Nov 05 2018

Crossrefs

Cf. A269804, A269806, A269807, A269808, A269809 (numbers with harmonic fractility 1, 3, ..., 6), A270000 (harmonic fractility of n).

Sequence in context: A045775 A356106 A178892 * A206715 A131853 A115817

Adjacent sequences: A269802 A269803 A269804 * A269806 A269807 A269808

Keyword

nonn,easy

Author

Clark Kimberling and Peter J. C. Moses, Mar 05 2016

Extensions

Edited by M. F. Hasler, Nov 05 2018

Status

approved