A270000 -id:A270000 - cp's OEIS frontend

A269804 Numbers having harmonic fractility 1, cf. A270000.

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

A269807 Numbers having harmonic fractility A270000(n) = 4.

Original entry on oeis.org

41, 71, 82, 109, 123, 141, 142, 157, 163, 164, 169, 175, 179, 181, 187, 191, 197, 211, 218, 229, 246, 251, 257, 265, 271, 282, 284, 293, 305, 311, 314, 323, 326, 327, 328, 338, 341, 350, 358, 362, 369, 371, 374, 382, 394, 395, 415, 422, 423, 433, 436, 445, 449

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 = 41:
NI(1/41) = (41, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...),
NI(2/41) = (20, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, ...),
NI(3/41) = (13, 3, 1, 1, 4, 2, 2, 20, 1, 1, 1, 2, 2, 1, 1, 1, 2, ...) ~ NI(2/41),
NI(4/41) = (10, 1, 2, 1, 1, 8, 1, 1, 8, 1, 1, 8, 1, 1, 8, 1, 1, 8, 1, 1, ...),
NI(5/41) = (6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, ...).
Any further NI(k/41) is equivalent to one of the  above, e.g., NI(40/11) = (1, 1, 1, 1, 1, 4, 2, 2, 20, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, ...) ~ NI(2/41).
Thus, the number of equivalence classes is 4 (represented by 1/41, 2/41, 4/41 and 5/41), so that the fractility of 41 is 4.

Jack W Grahl, Table of n, a(n) for n = 1..138

Cf. A269804, A269805, A269806, A269808, A269809 (numbers with harmonic fractility 1, 2, ..., 6), A270000 (harmonic fractility of n).

select(is_A269807(n)=A270000(n)==4, [1..450]) \\ M. F. Hasler, Nov 05 2018

More terms from Jack W Grahl, Jun 27 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

A269982 Factorial fractility of n.

Original entry on oeis.org

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

Offset: 2

Clark Kimberling and Peter J. C. Moses, Mar 10 2016

nonn

In order to define (factorial) 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 factorial fractility.
For factorial fractility, r(n) = 1/n!, n(j+1) = A084558(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(1/10) = (3, 2, 1, 1, 1, 2, 3, 2, 1, 1, 1, 2, 3, 2, 1, 1, 1, ...)
NI(2/10) = (2, 3, 2, 1, 1, 1, 2, 3, 2, 1, 1, 1, 2, 3, 2, 1, 1, ...)
NI(3/10) = (2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, ...)
NI(4/10) = (2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...)
NI(5/10) = (2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...)
NI(6/10) = (1, 2, 3, 2, 1, 1, 1, 2, 3, 2, 1, 1, 1, 2, 3, 2, 1, ...)
NI(7/10) = (1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, ...)
NI(8/10) = (1, 1, 2, 3, 2, 1, 1, 1, 2, 3, 2, 1, 1, 1, 2, 3, 2, ...)
NI(9/10) = (1, 1, 1, 2, 3, 2, 1, 1, 1, 2, 3, 2, 1, 1, 1, 2, 3, ...),
so that there are 3 equivalence classes for n = 10, and the factorial fractility of 10 is 3.

Robert Price, Table of n, a(n) for n = 2..500

Cf. A000142 (factorial numbers), A084558 (largest m: m! < n).
Cf. A269983, A269984, A269985, A269986, A269987, A269988: numbers with factorial fractility k = 1, 2, ..., 6, respectively.
Cf. A269570 (binary fractility), A270000 (harmonic fractility).

A269982[n_] := CountDistinct[With[{l = NestWhileList[Rescale[#, {1/(Floor[x] + 1)!, 1/Floor[x]!} /. FindRoot[1/x! == #, {x, 1}]] &, #, UnsameQ, All]}, Min@l[[First@First@Position[l, Last@l] ;;]]] & /@ Range[1/n, 1 - 1/n, 1/n]] (* Davin Park, Nov 19 2016 *)

A269982(n)=#Set(vector(n-1, k, NIFR(k/n))) \\ where:
NIFR(x, n, L=1, S=[], c=0)={for(i=2, oo, n=A084558(L\x); S=setunion(S, [x/L]); x-=L/(n+1)!; L/=(n+1)!\n; setsearch(S, x/L)&& if(c, break, c=!S=[])); S[1]} \\ variant of the function NIF() below; returns just a unique representative (smallest x/L occurring within the period) of the equivalence class.
NIF(x, n=[], L=1, S=[], c=0)={for(i=2, oo, n=concat(n, A084558(L\x)); c|| S=setunion(S, [x/L]); x-=L/(n[#n]+1)!; L/=(n[#n]-1)!*(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 [transition, period] of "factorial" NI(x). (End)

Edited by M. F. Hasler, Nov 05 2018

A269984 Numbers k having factorial fractility A269982(k) = 2.

Original entry on oeis.org

4, 5, 8, 9, 12, 14, 16, 18, 22, 23, 24, 26, 27, 32, 33, 37, 38, 39, 48, 49, 53, 54, 57, 58, 61, 64, 66, 78, 81, 83, 86, 87, 96, 97, 101, 107, 113, 114, 121, 129, 131, 139, 163, 169, 174, 178, 181, 193, 218, 227, 241, 257, 263, 267, 277, 302, 317, 327, 331

Offset: 1

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

nonn

See A269982 for a definition of factorial fractility and a guide to related sequences.

			NI(1/5) = (2, 3, 2, 1, 1, 1, 2, 3, 2, 1, 1, 1, 2, 3, 2, ...)
NI(2/5) = (2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...)
NI(3/5) = (1, 2, 3, 2, 1, 1, 1, 2, 3, 2, 1, 1, 1, 2, 3, ...)
NI(4/5) = (1, 1, 2, 3, 2, 1, 1, 1, 2, 3, 2, 1, 1, 1, 2, ...)
so there are 2 equivalences classes for n = 5, and the fractility of 5 is 2.

Robert Price, Table of n, a(n) for n = 1..67

Cf. A000142 (factorial numbers), A269982 (factorial fractility of n); A269983, A269985, A269986, A269987, A269988 (numbers with factorial fractility 1, 3, ..., 6, respectively).

Cf. A269570 (binary fractility), A270000 (harmonic fractility).

A269982[n_] := CountDistinct[With[{l = NestWhileList[
         Rescale[#, {1/(Floor[x] + 1)!, 1/Floor[x]!} /.
            FindRoot[1/x! == #, {x, 1}]] &, #, UnsameQ, All]},
      Min@l[[First@First@Position[l, Last@l] ;;]]] & /@
    Range[1/n, 1 - 1/n, 1/n]]; (* Davin Park, Nov 19 2016 *)
Select[Range[2, 500], A269982[#] == 2 &] (* Robert Price, Sep 19 2019 *)

select( is_A269984(n)=A269982(n)==2, [1..300]) \\ M. F. Hasler, Nov 05 2018

Edited by M. F. Hasler, Nov 05 2018

A269985 Numbers k having factorial fractility A269982(k) = 3.

Original entry on oeis.org

10, 15, 17, 21, 25, 30, 36, 41, 42, 44, 52, 55, 62, 72, 74, 76, 88, 93, 98, 99, 103, 104, 106, 108, 111, 118, 122, 125, 128, 132, 134, 137, 146, 149, 155, 158, 162, 166, 173, 176, 177, 179, 183, 186, 192, 198, 201, 202, 203, 214, 219, 226, 228, 237, 242, 249

Offset: 1

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

nonn

See A269982 for a definition of factorial fractility and a guide to related sequences.

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

Robert Price, Table of n, a(n) for n = 1..91

Cf. A000142 (factorial numbers), A269982 (factorial fractility of n); A269983, A269984, A269986, A269987, A269988 (numbers with factorial fractility 1, 2, ..., 6, respectively).

Cf. A269570 (binary fractility), A270000 (harmonic fractility).

A269982[n_] := CountDistinct[With[{l = NestWhileList[
        Rescale[#, {1/(Floor[x] + 1)!, 1/Floor[x]!} /.
           FindRoot[1/x! == #, {x, 1}]] &, #, UnsameQ, All]},
     Min@l[[First@First@Position[l, Last@l] ;;]]] & /@
   Range[1/n, 1 - 1/n, 1/n]]; (* Davin Park, Nov 19 2016 *)
Select[Range[2, 500], A269982[#] == 3 &] (* Robert Price, Sep 19 2019 *)

select( is_A269985(n)=A269982(n)==2, [1..200]) \\ M. F. Hasler, Nov 05 2018

Edited by M. F. Hasler, Nov 05 2018

A269986 Numbers k having factorial fractility A269982(k) = 4.

Original entry on oeis.org

20, 28, 34, 35, 40, 45, 46, 47, 50, 51, 56, 60, 63, 65, 69, 75, 77, 80, 82, 84, 90, 91, 102, 110, 112, 116, 117, 120, 123, 124, 133, 135, 144, 147, 148, 150, 152, 156, 159, 160, 165, 167, 171, 172, 194, 206, 208, 209, 216, 217, 222, 223, 234, 236, 239, 240

Offset: 1

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

nonn

See A269982 for a definition of factorial fractility and a guide to related sequences.

			NI(1/20) = (3, 3, 2, 3, 2, 1, 1, 1, 2, 3, 2, 1, 1, 1, 2, 3, 2, ...)
NI(5/20) = (2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...)
NI(6/20) = (2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, ...)
NI(10/20) = (2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...).
These 4 equivalence classes represent all the classes for n = 20, so the factorial fractility of 20 is 4.

Robert Price, Table of n, a(n) for n = 1..116

Cf. A000142 (factorial numbers), A269982 (factorial fractility of n); A269983, A269984, A269985, A269987, A269988 (numbers with factorial fractility 1, 2, ..., 6, respectively).

Cf. A269570 (binary fractility), A270000 (harmonic fractility).

A269982[n_] := CountDistinct[With[{l = NestWhileList[
        Rescale[#, {1/(Floor[x] + 1)!, 1/Floor[x]!} /.
           FindRoot[1/x! == #, {x, 1}]] &, #, UnsameQ, All]},
     Min@l[[First@First@Position[l, Last@l] ;;]]] & /@
   Range[1/n, 1 - 1/n, 1/n]]; (* Davin Park, Nov 19 2016 *)
Select[Range[2, 500], A269982[#] == 4 &] (* Robert Price, Sep 19 2019 *)

select( is_A269986(n)=A269982(n)==4, [1..200]) \\ M. F. Hasler, Nov 05 2018

Edited by M. F. Hasler, Nov 05 2018

cp's OEIS Frontend

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