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Number of k >= 1 with k! == 1 (mod n).
5

%I #34 Jun 26 2019 12:32:28

%S 0,1,1,1,2,1,2,1,1,1,2,1,2,1,1,1,4,1,2,1,1,1,5,1,1,1,1,1,3,1,3,1,1,1,

%T 1,1,2,1,1,1,2,1,2,1,1,1,2,1,1,1,1,1,4,1,1,1,1,1,4,1,5,1,1,1,1,1,3,1,

%U 1,1,3,1,4,1,1,1,1,1,2,1,1,1,4,1,1,1,1,1,4,1,1,1,1,1,1,1,4,1,1,1,2,1,3,1,1

%N Number of k >= 1 with k! == 1 (mod n).

%C The first occurrences for 0..10 are 1, 2, 5, 29, 17, 23, 199, 619, 3313, 4093, 3011, ... (see A049050). - _Antti Karttunen_, Oct 01 2018

%H Antti Karttunen, <a href="/A049046/b049046.txt">Table of n, a(n) for n = 1..12021</a>

%e From _Antti Karttunen_, Oct 01 2018: (Start)

%e a(1) = 0 because 1 divides all factorial numbers (A000142): 1, 2, 6, 24, ... and thus there are no cases where the remainder would be 1.

%e a(3) = 1 as (1! mod 3) = 1, (2! mod 3) = 2 and for 3! and larger factorials the remainder is always 0. Thus there is exactly one case where the remainder is one.

%e a(5) = 2 as (1! mod 5) = 1, (2! mod 5) = 2, (3! mod 5) = 1, (4! mod 5) = 5, (5! mod 5) = 0, and so on ever after for larger factorials.

%e (End)

%t Table[Length[Select[Range[100], Mod[#!, n] == 1 &]], {n, 1, 100}] (* _G. C. Greubel_, Oct 08 2018 *)

%o (PARI) A049046(n) = { my(s=0, r, k=1); while((r=(k! % n))>0, s += (1==r); k++); (s); }; \\ _Antti Karttunen_, Oct 01 2018

%Y Cf. A000142, A002034, A071710, A038203, A049050.

%K nonn

%O 1,5

%A _David W. Wilson_

%E Term a(1) corrected and the definition clarified by _Antti Karttunen_, Oct 01 2018

%E Definition further edited by _Antti Karttunen_, Oct 06 2018, based on feedback from _David W. Wilson_