A269807 -id:A269807 - cp's OEIS frontend

A270000 Harmonic fractility of n.

1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 3, 1, 2, 1, 1, 1, 3, 2, 1, 3, 3, 1, 3, 3, 1, 1, 3, 2, 1, 1, 3, 1, 3, 1, 2, 3, 3, 2, 4, 1, 2, 3, 2, 3, 3, 1, 1, 3, 1, 3, 3, 1, 5, 1, 3, 3, 2, 2, 2, 1, 1, 1, 5, 3, 3, 1, 3, 3, 4, 1, 2, 2, 3, 3, 6, 3, 3, 2, 1, 4, 3, 1, 2, 2, 3, 3

Offset: 2

Clark Kimberling and Peter J. C. Moses, Mar 11 2016

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 index n such that r(n(1)+1) + L(1)*r(n+1) < 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 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.
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

			NI(1/11) = (11, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...),
NI(2/11) = (5, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, ...),
NI(3/11) = (3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, ...),
NI(4/11) = (2, 5, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...),
NI(5/11) = (2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...),
NI(6/11) = (1, 11, 1, 1, 1, 1, 1, 1, 1, 1, ...),
NI(7/11) = (1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, ...),
NI(8/11) = (1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, ...),
NI(9/11) = (1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, ...),
NI(10/11) = (1, 1, 1, 3, 3, 3, 3, 3, 3, 3, ...),
so that there are 3 equivalence classes for n = 11, and that the harmonic fractility of 11 is 3.

Jack W Grahl, Table of n, a(n) for n = 2..999
Jack W Grahl, Python code to generate this sequence

Guide to related sequences:
  k - numbers with harmonic fractility k:
  1 - A269804
  2 - A269805
  3 - A269806
  4 - A269807
  5 - A269808
  6 - A269809
Cf. A269570 (binary fractility), A269982 (factorial fractility).

A270000[n_] := CountDistinct[With[{l = NestWhileList[Rescale[#, {1/(Floor[1/#] + 1), 1/Floor[1/#]}] &, #, UnsameQ, All]}, Min@l[[First@FirstPosition[l, Last@l] ;;]]] & /@ Range[1/n, 1 - 1/n, 1/n]] (* Davin Park, Nov 09 2016 *)

A270000(n)=#Set(vector(n-1,k,NIR(k/n))) \\ where:
NIR(x, n, L=1, S=[], c=0)={for(i=2, oo, n=L\x; S=setunion(S, [x/L]); x-=L/(n+1); L/=n*(n+1); setsearch(S, x/L)&& if(c, break, c=!S=[])); S[1]} \\ variant of the function NI() below; returns just a unique representative (smallest x/L occurring within the period) of the equivalence class.
NI(x, n=[], L=1, S=[], c=0)={for(i=2, oo, n=concat(n, L\x); c|| S=setunion(S, [x/L]); x-=L/(n[#n]+1); L/=n[#n]*(n[#n]+1); if(!c, setsearch(S, x/L)&& [c,S]=[i,x/L], x/L==S, c-=i; break)); [n[1..2*c-1], n[c..-1]]} \\ Returns the harmonic nested interval sequence for x in the form [transition, period]. (End)

Definition corrected by Jack W Grahl, Jun 27 2018

Edited by M. F. Hasler, Nov 05 2018

A269804 Numbers having harmonic fractility 1, cf. A270000.

Original entry on oeis.org

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, 56, 62, 63, 64, 68, 72, 81, 84, 93, 96, 98, 102, 108, 112, 113, 124, 126, 128, 136, 144, 147, 151, 153, 162, 168, 186, 189, 192, 196, 204, 216, 224, 226, 241, 243

Offset: 1

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

nonn

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.
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.
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

			Nested-interval sequences NI(k/m) for m = 6:
NI(1/6) = (6, 1, 1, 1, 1, 1,...)
NI(2/6) = (3, 1, 1, 1, 1, 1,...)
NI(3/6) = (2, 1, 1, 1, 1, 1,...)
NI(4/6) = (1, 3, 1, 1, 1, 1,...)
NI(5/6) = (1, 1, 3, 1, 1, 1,...):
There is only one equivalence class, so that the fractility of 6 is 1.

Jack W Grahl, Table of n, a(n) for n = 1..117
Peter J. C. Moses, Clark Kimberling, Nested interval sequences of positive real numbers, Integers 17 (2017), #A46.

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

Mathematica
```
(* see A270000 *)
```

select( is_A269804(n)=A270000(n)==1, [1..250]) \\ M. F. Hasler, Nov 05 2018

Edited by M. F. Hasler, Nov 05 2018

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

Original entry on oeis.org

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

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

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.

			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.

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

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

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

Edited by M. F. Hasler, Nov 05 2018

A269806 Numbers having harmonic fractility A270000(n) = 3.

Original entry on oeis.org

11, 13, 19, 22, 23, 25, 26, 29, 33, 35, 38, 39, 44, 46, 47, 50, 52, 53, 57, 58, 66, 67, 69, 70, 75, 76, 78, 79, 83, 87, 88, 89, 92, 94, 99, 100, 104, 105, 106, 114, 116, 117, 119, 125, 132, 133, 134, 138, 140, 149, 150, 152, 155, 156, 158, 159, 161, 166, 171

Offset: 1

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

nonn

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.
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.
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

			Nested interval sequences NI(k/m) for m = 11:
NI(1/11) = (11,1, 1, 1, 1, 1, 1, 1, ...),
NI(2/11) = (5, 2, 1, 2, 1, 2, 1, 1, 2, ...),
NI(3/11) = (3, 3, 3, 3, 3, 3, 3, 3, 3, ...),
NI(4/11) = (2, 5, 2, 1, 2, 1, 2, 1, 2, ...),
NI(5/11) = (2, 1, 2, 1, 2, 1, 2, 1, 2, ...) equivalent to NI(4/11),
NI(6/11) = (1, 11, 1, 1, 1, 1, 1, 1, ...) equivalent to NI(1/11),
NI(7/11) = (1, 3, 3, 3, 3, 3, 3, 3, 3, ...) equivalent to NI(3/11),
NI(8/11) = (1, 2, 1, 2, 1, 2, 1, 2, 1, ...) equivalent to NI(4/11),
NI(9/11) = (1, 1, 3, 3, 3, 3, 3, 3, 3, ...) equivalent to NI(3/11),
NI(10/11) = (1, 1, 1, 3, 3, 3, 3, 3, ...) equivalent to NI(3/11).
So there are 3 equivalence classes for m = 11, and the fractility of 11 is 3.

Jack W Grahl, Table of n, a(n) for n = 1..299

Cf. A269804, A269805, A269807, A269808, A269809 (numbers with harmonic fractility 1, 2, 4, 5, 6, respectively); A270000 (harmonic fractility of n).

select( is_A269806(n)=A270000(n)==3, [1..300]) \\ M. F. Hasler, Nov 05 2018

Edited by M. F. Hasler, Nov 05 2018

A269808 Numbers having harmonic fractility A270000(n) = 5.

Original entry on oeis.org

55, 65, 91, 110, 115, 121, 130, 137, 165, 182, 195, 205, 213, 220, 221, 230, 235, 242, 260, 273, 274, 295, 330, 335, 337, 345, 355, 361, 363, 364, 390, 391, 403, 407, 410, 411, 419, 426, 440, 442, 460, 467, 470, 481, 484, 485, 495, 497, 503, 505, 517, 520, 546

Offset: 1

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

nonn

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.
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.
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

			Nested interval sequences NI(k/m) for m = 55:
The 5 equivalence classes are represented by
NI(1/55) = (55, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...),
NI(2/55) = (27, 2, 1, 1, 1, 1, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, ...),
NI(4/55) = (13, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1,...),
NI(6/55) = (9, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, ...),
NI(22/55) = (2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...).
For example, NI(3/55) = (18, 1, 3, 1, 2, 1, 55, 1, 1, 1, ...) is equivalent to NI(1/55).

Jack W Grahl, Table of n, a(n) for n = 1..115

Cf. A269804, A269805, A269806, A269807, A269809 (numbers with harmonic fractility 1, 2, ..., 6), A270000 (harmonic fractility of n).

select( is_A269808(n)=A270000(n)==5, [1..550]) \\ M. F. Hasler, Nov 05 2018

More terms from Jack W Grahl, Jun 27 2018

Edited by M. F. Hasler, Nov 05 2018

A269809 Numbers having harmonic fractility A270000(n) = 6.

Original entry on oeis.org

77, 95, 131, 145, 154, 190, 203, 209, 231, 247, 262, 275, 285, 290, 299, 308, 329, 377, 380, 393, 406, 418, 431, 435, 437, 443, 462, 494, 524, 529, 539, 545, 550, 559, 570, 580, 595, 598, 609, 616, 627, 658, 685, 689, 693, 705, 737, 741, 754, 760, 767, 786

Offset: 1

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

nonn

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.
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.
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

			Nested interval sequences NI(k/m) for m = 77:
The 6 equivalence classes are represented by
NI(1/77) = (77, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...),
NI(2/77) = (38, 2, 1, 1, 1, 1, 2, 3, 8, 2, 2, 1, 1, 1, 1, 2, 3, 8, 2, 2, 1, ...) (period length 9),
NI(3/77) = (25, 3, 1, 1, 1, 5, 15, 1, 5, 15, 1, 5, 15, 1, 5, 15, 1, 5, 15, ...),
NI(8/77) = (9, 2, 9, 2, 9, 2, 9, 2, 9, 2, 9, 2, 9, 2, 9, 2, 9, 2, 9, 2, 9, ...),
NI(10/77) = (7, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, ...),
NI(14/77) = (5, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, ...).
For example, N(4/77) = (19, 1, 2, 1, 1, 1, 15, 1, 5, 15, 1, 5, ...) is equivalent to NI(3/77), and NI(6/77) = (12, 6, 1, 11, 1, 1, 1, ...) is equivalent to NI(1/77). - _M. F. Hasler_, Nov 05 2018

Jack W Grahl, Table of n, a(n) for n = 1..73
Peter J. C. Moses, Clark Kimberling, Nested interval sequences of positive real numbers, Integers 17 (2017), #A46.

Cf. A269804, A269805, A269806, A269807, A269808 (numbers with harmonic fractility 1, 2, 3, 4, 5, respectively); A270000 (harmonic fractility of n).

select( is_A269809(n)=A270000(n)==6, [1..800]) \\ M. F. Hasler, Nov 05 2018

More terms from Jack W Grahl, Jun 28 2018

Edited by M. F. Hasler, Nov 05 2018

cp's OEIS Frontend

A270000 Harmonic fractility of n.

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A269804 Numbers having harmonic fractility 1, cf. A270000.

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

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A269806 Numbers having harmonic fractility A270000(n) = 3.

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A269808 Numbers having harmonic fractility A270000(n) = 5.

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A269809 Numbers having harmonic fractility A270000(n) = 6.

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