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%I #20 Oct 06 2023 03:28:36
%S 10,15,17,21,25,30,36,41,42,44,52,55,62,72,74,76,88,93,98,99,103,104,
%T 106,108,111,118,122,125,128,132,134,137,146,149,155,158,162,166,173,
%U 176,177,179,183,186,192,198,201,202,203,214,219,226,228,237,242,249
%N Numbers k having factorial fractility A269982(k) = 3.
%C See A269982 for a definition of factorial fractility and a guide to related sequences.
%H Robert Price, <a href="/A269985/b269985.txt">Table of n, a(n) for n = 1..91</a>
%e NI(1/10) = (3, 2, 1, 1, 1, 2, 3, 2, 1, 1, 1, 2, 3, 2, 1, 1, 1, ...),
%e NI(2/10) = (2, 3, 2, 1, 1, 1, 2, 3, 2, 1, 1, 1, 2, 3, 2, 1, 1, ...) ~ NI(1/10),
%e NI(3/10) = (2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, ...),
%e NI(4/10) = (2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...) ~ NI(3/10),
%e NI(5/10) = (2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...),
%e NI(6/10) = (1, 2, 3, 2, 1, 1, 1, 2, 3, 2, 1, 1, 1, 2, 3, 2, 1, ...) ~ NI(1/10),
%e NI(7/10) = (1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, ...) ~ NI(3/10),
%e NI(8/10) = (1, 1, 2, 3, 2, 1, 1, 1, 2, 3, 2, 1, 1, 1, 2, 3, 2, ...) ~ NI(1/10),
%e NI(9/10) = (1, 1, 1, 2, 3, 2, 1, 1, 1, 2, 3, 2, 1, 1, 1, 2, 3, ...) ~ NI(1/10),
%e so that there are 3 equivalence classes for n = 10, so the factorial fractility of 10 is 3.
%t A269982[n_] := CountDistinct[With[{l = NestWhileList[
%t Rescale[#, {1/(Floor[x] + 1)!, 1/Floor[x]!} /.
%t FindRoot[1/x! == #, {x, 1}]] &, #, UnsameQ, All]},
%t Min@l[[First@First@Position[l, Last@l] ;;]]] & /@
%t Range[1/n, 1 - 1/n, 1/n]]; (* _Davin Park_, Nov 19 2016 *)
%t Select[Range[2, 500], A269982[#] == 3 &] (* _Robert Price_, Sep 19 2019 *)
%o (PARI) select( is_A269985(n)=A269982(n)==2, [1..200]) \\ _M. F. Hasler_, Nov 05 2018
%Y Cf. A000142 (factorial numbers), A269982 (factorial fractility of n); A269983, A269984, A269986, A269987, A269988 (numbers with factorial fractility 1, 2, ..., 6, respectively).
%Y Cf. A269570 (binary fractility), A270000 (harmonic fractility).
%K nonn
%O 1,1
%A _Clark Kimberling_ and _Peter J. C. Moses_, Mar 11 2016
%E Edited by _M. F. Hasler_, Nov 05 2018
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