A270000 -id:A270000 - cp's OEIS frontend

A269987 Numbers k having factorial fractility A269982(k) = 5.

68, 70, 71, 85, 92, 100, 126, 127, 130, 136, 138, 145, 154, 157, 161, 164, 168, 180, 185, 195, 200, 204, 220, 224, 232, 247, 253, 266, 272, 288, 291, 300, 304, 310, 318, 324, 328, 333, 334, 336, 341, 342, 348, 360, 365, 369, 371, 390, 395, 400, 404, 407, 408, 412, 418, 433, 440, 441, 443, 444, 447

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(1/68) = (4, 2, 3, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, ...)
NI(4/68) = (3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, ...)
NI(6/68) = (3, 2, 1, 2, 2, 3, 1, 2, 3, 1, 1, 2, 1, 2, 2, 3, 1, 2, 3, ...)
NI(17/68) = (2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...)
NI(34/68) = (2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...).
These 5 equivalence classes represent all the classes for n = 68, so the factorial fractility of 68 is 5.

Robert Price, Table of n, a(n) for n = 1..70

Cf. A000142 (factorial numbers), A269982 (factorial fractility of n); A269983, A269984, A269985, A269986, 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[#] == 5 &] (* Robert Price, Sep 19 2019 *)

select( is_A269987(n)=A269982(n)==5, [1..400]) \\ M. F. Hasler, Nov 05 2018

Edited and more terms added by M. F. Hasler, Nov 05 2018

A269988 Numbers k having factorial fractility A269982(k) = 6.

Original entry on oeis.org

94, 105, 115, 141, 142, 153, 170, 175, 182, 184, 187, 189, 196, 205, 207, 210, 212, 213, 215, 221, 225, 235, 245, 252, 254, 255, 260, 265, 275, 276, 282, 290, 299, 306, 314, 325, 367, 368, 370, 378, 381, 388, 392, 399, 414, 424, 425, 426, 434, 435, 446, 450

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(1/94) = (4, 3, 2, 3, 1, 1, 1, 1, 3, 2, 1, 2, 1, 1, 3, 1, 4, 1, 1, 2, ...),
NI(2/94) = (4, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, ...),
NI(4/94) = (3, 5, 1, 1, 2, 2, 1, 3, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 3, ...),
NI(7/94) = (3, 2, 2, 2, 1, 2, 4, 3, 2, 3, 1, 1, 1, 1, 3, 2, 1, 2, 1, 1, ...),
NI(11/94) = (3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, ...),
NI(47/94) = (2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...):
These 6 equivalence classes represent all the classes for k = 94, so the factorial fractility of 94 is 6.

Robert Price, Table of n, a(n) for n = 1..60

Cf. A000142 (factorial numbers), A269982 (factorial fractility of n); A269983, A269984, A269985, A269986, A269987 (numbers with factorial fractility 1, 2, ..., 5, 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[#] == 6 &] (* Robert Price, Sep 19 2019 *)

select( is_A269988(n)=A269982(n)==6, [1..400]) \\ M. F. Hasler, Nov 05 2018

Edited and more terms added by M. F. Hasler, Nov 05 2018

A269983 Numbers k having factorial fractility A269982(k) = 1.

Original entry on oeis.org

2, 3, 6, 7, 11, 13, 19, 29, 31, 43, 59, 67, 73, 79, 89, 109, 151, 197, 199, 211, 229, 233, 269, 281, 283, 293, 337, 373, 379, 389, 397, 419, 421, 439, 449, 463, 487, 503, 509, 547, 557, 619, 673, 701, 727, 733, 797, 809, 811, 827, 877, 883, 887, 937, 941, 947, 953, 983

Offset: 1

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

nonn

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

Is 6 the largest even term of this sequence? - M. F. Hasler, Nov 05 2018

			NI(1/7) = (3, 1, 1, 2, 2, 3, 1, 1, 2, 2, 3, 1, 1, 2, 2, ...),
NI(2/7) = (2, 2, 1, 3, 1, 1, 2, 2, 3, 1, 1, 2, 2, 3, 1, ...),
NI(3/7) = (2, 1, 1, 3, 1, 1, 2, 2, 3, 1, 1, 2, 2, 3, 1, ...),
NI(4/7) = (1, 3, 1, 1, 2, 2, 3, 1, 1, 2, 2, 3, 1, 1, 2, ...),
NI(5/7) = (1, 2, 1, 1, 3, 1, 1, 2, 2, 3, 1, 1, 2, 2, 3, ...),
NI(6/7) = (1, 1, 2, 1, 1, 3, 1, 1, 2, 2, 3, 1, 1, 2, 2, ...):
all are eventually periodic with period (1, 1, 2, 2, 3), so there is only one equivalence class for n = 7, and the fractility of 7 is 1.

Cf. A269982 (factorial fractility of n); A269984, A269985, A269986, A269987, A269988 (numbers with factorial fractility 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, 1000], A269982[#] == 1 &] (* Robert Price, Sep 19 2019 *)

select( is_A269983(n)=A269982(n)==1, [1..300]) \\ M. F. Hasler, Nov 05 2018

Edited and more terms added by M. F. Hasler, Nov 05 2018

a(54)-a(58) from Robert Price, Sep 19 2019

A269989 Odds fractility of n.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 3, 3, 6, 6, 5, 6, 7, 6, 8, 7, 5, 8, 8, 8, 10, 10, 10, 10, 11, 5, 13, 11, 9, 15, 13, 11, 14, 15, 10, 16, 15, 11, 15, 18, 14, 18, 18, 10, 23, 17, 14, 18, 15, 16, 25, 20, 10, 20, 24, 15, 25, 23, 16, 27, 27, 14, 23, 24, 21, 26, 27, 25, 26, 29

Offset: 2

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

In order to define (odds) 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 = (1/1, 1/3, 1/5, 1/7, 1/9, ... ) gives odds fractilily.
binary fractility:  A269570
factorial fractility:  A269982
harmonic fractility:  A270000
primes fractility:  A269990

			NI(1/7) = (4,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...)
NI(2/7) = (2,1,1,3,1,1,1,1,1,1,2,1,1,2,2,1,2,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,2,1,2,1,1,1,19,1,30,1,2,2,1,10,1,1,3,1,...)
NI(3/7) = (1,4,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...)
NI(4/7) = (1,1,14,1,2,3,1,2,1,1,1,1,1,2,1,3,2,1,6,1,1,11,1,1,1,1,1,1,1,12,2,1,1,1,2,3,1,1,1,1,1,6,1,1,1,1,2,3,1,7,...)
NI(5/7) = (1,1,1,14,1,2,3,1,2,1,1,1,1,1,2,1,3,2,1,6,1,1,11,1,1,1,1,1,1,1,12,2,1,1,1,2,3,1,1,1,1,1,6,1,1,1,1,2,3,1,...)
NI(6/7) = (1,1,1,1,2,1,1,11,1,2,1,1,1,1,1,1,1,6,1,7,1,1,1,1,1,1,1,2,1,1,6,1,1,1,194,1,2,7,6,2,1,1,1,1,1,1,3,1,2,1,...);
there are 4 equivalence classes:  {1/7,3/7},{2/7},{4,5},{6},so that a(7) = 4.

Cf. A005408, A269570, A269982, A270000, A269990.

A269990 Primes fractility of n.

Original entry on oeis.org

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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A269987 Numbers k having factorial fractility A269982(k) = 5.

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A269988 Numbers k having factorial fractility A269982(k) = 6.

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A269983 Numbers k having factorial fractility A269982(k) = 1.

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A269989 Odds fractility of n.

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A269990 Primes fractility of n.

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