A269989 -id:A269989 - cp's OEIS frontend

A269990 Primes fractility of n.

1, 2, 2, 3, 4, 4, 4, 5, 6, 6, 6, 7, 8, 8, 9, 9, 6, 10, 10, 10, 12, 12, 11, 12, 15, 7, 14, 15, 12, 17, 16, 13, 17, 15, 13, 18, 18, 16, 18, 23, 16, 20, 21, 14, 22, 23, 19, 23, 22, 20, 27, 26, 16, 24, 26, 21, 28, 27, 20, 29, 32, 18, 30, 33, 27, 35, 33, 27, 29

Offset: 2

Clark Kimberling and Peter J. C. Moses, Mar 12 2016

In order to define (primes) 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 index n such that r(n(1)+1) < x <= r(n(1)+1) + L(1)r(n), and let L(2) = (r(n(2)) - r(r(n)+1))L(1). Continue inductively to obtain the sequence (n(1), n(2), n(3), ...), the r-nested interval sequence of x.
For fixed r, call x and y equivalent if NI(x) and NI(y) are eventually identical. 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 = (1/2, 1/3, 1/5, 1/7, 1/11, ... ) gives primes fractility.
binary fractility: A269570
factorial fractility: A269982
harmonic fractility: A270000
odds fractility: A269989

			NI(1/11) = (5,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...)
NI(2/11) = (3,1,1,2,1,1,3,1,3,1,1,1,2,2,1,1,1,2,1,1,1,4,4,1,2,10,1,1,1,1,1,1,1,1,1,2,1,1,8,1,1,1,1,1,2,1,2,1,1,1,1,1,2,1,4,1,1,3,1,8,1,1,1,1,1,1,...)
NI(3/11) = (2,1,2,1,1,1,1,1,4,1,1,1,1,1,3,2,9,1,1,1,2,1,2,2,2,2,1,1,1,4,1,1,1,1,1,1,1,1,1,1,11,1,2,4,1,4,3,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,2,...)
NI(4/11) = (1,8,2,4,1,2,1,1,1,1,1,2,1,3,2,1,5,1,1,8,1,4,1,1,1,1,1,1,1,2,3,3,1,3,1,1,1,1,1,5,2,3,2,4,2,1,8,2,1,1,2,2,106,2,3,1,1,1,1,1,1,2,2,6,1,,...)
NI(5/11) = (1,3,1,1,2,1,1,3,1,3,1,1,1,2,2,1,1,1,2,1,1,1,4,4,1,2,10,1,1,1,1,1,1,1,1,1,2,1,1,8,1,1,1,1,1,2,1,2,1,1,1,1,1,2,1,4,1,1,3,1,8,1,1,1,1,1,1,...)
NI(6/11) = (1,2,1,1,1,1,1,4,1,1,1,1,1,3,2,9,1,1,1,2,1,2,2,2,2,1,1,1,4,1,1,1,1,1,1,1,1,1,1,11,1,2,4,1,4,3,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,2,1,1,1,...);
NI(7/11) = (1,1,3,1,1,2,1,1,3,1,3,1,1,1,2,2,1,1,1,2,1,1,1,4,4,1,2,10,1,1,1,1,1,1,1,1,1,2,1,1,8,1,1,1,1,1,2,1,2,1,1,1,1,1,2,1,4,1,1,3,1,8,1,1,1,1,1,1,...);
NI(8/11) = (1,1,1,5,3,1,1,1,2,1,3,1,1,1,1,1,2,1,11,1,1,1,1,1,1,1,1,2,1,1,1,2,1,8,1,1,2,3,1,1,1,6,1,2,1,4,1,1,1,1,1,1,34,1,8,1,3,1,1,5,1,1,1,1,1,4,1,...);
NI(9/11) = (1,1,1,1,5,3,1,1,1,2,1,3,1,1,1,1,1,2,1,11,1,1,1,1,1,1,1,1,2,1,1,1,2,1,8,1,1,2,3,1,1,1,6,1,2,1,4,1,1,1,1,1,1,34,1,8,1,3,1,1,5,1,1,1,1,1,4,...);
NI(10/11) = (1,1,1,1,1,2,1,1,1,1,4,3,1,1,1,1,1,1,2,1,1,1,1,6,1,1,1,1,3,2,1,1,1,1,5,7,1,3,2,1,3,1,1,1,1,1,1,1,1,3,1,1,2,2,4,2,1,1,1,1,1,1,1,1,1,1,1,6,...);  there are 6 equivalence classes: {1/11}, {2/11,5/11,7/11},{3,11,6/11},{4/11},{8/11,9/11},{10/11}, so that a(11) = 6.

Cf. A000040, A269570, A269982, A269989, A270000.

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A269990 Primes fractility of n.

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