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%I #15 Oct 06 2023 03:28:15
%S 2,3,6,7,11,13,19,29,31,43,59,67,73,79,89,109,151,197,199,211,229,233,
%T 269,281,283,293,337,373,379,389,397,419,421,439,449,463,487,503,509,
%U 547,557,619,673,701,727,733,797,809,811,827,877,883,887,937,941,947,953,983
%N Numbers k having factorial fractility A269982(k) = 1.
%C See A269982 for a definition of factorial fractility and a guide to related sequences.
%C Is 6 the largest even term of this sequence? - _M. F. Hasler_, Nov 05 2018
%e NI(1/7) = (3, 1, 1, 2, 2, 3, 1, 1, 2, 2, 3, 1, 1, 2, 2, ...),
%e NI(2/7) = (2, 2, 1, 3, 1, 1, 2, 2, 3, 1, 1, 2, 2, 3, 1, ...),
%e NI(3/7) = (2, 1, 1, 3, 1, 1, 2, 2, 3, 1, 1, 2, 2, 3, 1, ...),
%e NI(4/7) = (1, 3, 1, 1, 2, 2, 3, 1, 1, 2, 2, 3, 1, 1, 2, ...),
%e NI(5/7) = (1, 2, 1, 1, 3, 1, 1, 2, 2, 3, 1, 1, 2, 2, 3, ...),
%e NI(6/7) = (1, 1, 2, 1, 1, 3, 1, 1, 2, 2, 3, 1, 1, 2, 2, ...):
%e 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.
%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, 1000], A269982[#] == 1 &] (* _Robert Price_, Sep 19 2019 *)
%o (PARI) select( is_A269983(n)=A269982(n)==1, [1..300]) \\ _M. F. Hasler_, Nov 05 2018
%Y Cf. A269982 (factorial fractility of n); A269984, A269985, A269986, A269987, A269988 (numbers with factorial fractility 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 10 2016
%E Edited and more terms added by _M. F. Hasler_, Nov 05 2018
%E a(54)-a(58) from _Robert Price_, Sep 19 2019
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