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

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

Author

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

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

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

Crossrefs

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).

Programs

Mathematica

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 *)

PARI

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

Extensions

Edited by M. F. Hasler, Nov 05 2018