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Numbers k such that Fibonacci(k) divides k!.
10

%I #26 Nov 09 2021 11:07:09

%S 1,2,3,4,5,6,8,12,24

%N Numbers k such that Fibonacci(k) divides k!.

%C These terms m are called “triphobe” or “3-phobe” numbers, by the French website Diophante (see link), because there are no 3 positive integers b_1 < b_2 < b_3 such that b_1 divides b_2, b_2 divides b_3, and m = b_1 + b_2 + b_3. A number that is not “triphobe” is called “triphile” or “3-phile” (A160811). The set of k-phobe numbers is always finite, there exist 9 triphobe numbers and the largest one is 24. - _Bernard Schott_, Oct 23 2021

%D Posting to math-fun(AT)cs.arizona.edu by R. W. Gosper Nov 06 1996.

%H Diophante, <a href="http://www.diophante.fr/problemes-par-themes/arithmetique-et-algebre/a4-equations-diophantiennes/3143-a496-pentaphiles-et-pentaphobes">A496 - Pentaphiles et pentaphobes</a> (in French).

%t Select[Range[30],Divisible[#!,Fibonacci[#]]&] (* _Harvey P. Dale_, Jun 14 2020 *)

%Y k-phobe numbers: this sequence (k=3), A348519 (k=4), A348520 (k=5).

%Y k-phile numbers: A160811 \ {5} (k=3), A348517 (k=4), A348518 (k=5).

%Y Cf. A000045, A000142, A160811.

%K nonn,fini,full

%O 1,2

%A _N. J. A. Sloane_.

%E Offset changed to 1 by _David A. Corneth_, Oct 27 2021