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

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

Author

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

Keywords

• nonn

Comments

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

Examples

			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.
		

Crossrefs

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).

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, 1000], A269982[#] == 1 &] (* Robert Price, Sep 19 2019 *)

PARI

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

Extensions

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

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