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A205561 Least positive integer k such that n divides (2k)! - (2j)! for some j in [1,k-1].

%I A205561 #15 Nov 20 2024 19:17:51
%S A205561 2,2,3,3,4,3,5,3,4,4,2,3,8,5,4,4,10,4,4,4,5,2,4,3,4,8,6,5,3,4,5,5,4,
%T A205561 10,5,4,20,4,8,4,11,5,10,4,4,4,5,4,6,4,10,8,6,6,4,5,8,3,13,4,8,5,5,5,
%U A205561 8,4,16,10,4,5,7,4,4,20,4,8,7,8,11,4,6,11,22,5,10,10,3,4,5,4,8,4
%N A205561 Least positive integer k such that n divides (2k)! - (2j)! for some j in [1,k-1].
%C A205561 For a guide to related sequences, see A204892.
%C A205561 From _Robert Israel_, Nov 20 2024: (Start)
%C A205561 a(n) <= ceil(A002034(n)/2) + 1.
%C A205561 The last occurrence of k >= 2 in the sequence is a((2*k)! - 2) = k. (End)
%H A205561 Robert Israel, <a href="/A205561/b205561.txt">Table of n, a(n) for n = 1..10000</a>
%e A205561 1 divides (2*2)!-(2*1)! -> k=2, j=1
%e A205561 2 divides (2*2)!-(2*1)! -> k=2, j=1
%e A205561 3 divides (2*3)!-(2*2)! -> k=3, j=2
%e A205561 4 divides (2*3)!-(2*2)! -> k=3, j=2
%e A205561 5 divides (2*4)!-(2*3)! -> k=4, j=3
%p A205561 f:= proc(n) local S,j,x;
%p A205561   S:= {}:
%p A205561   x:= 1:
%p A205561   for j from 1 do
%p A205561     x:=x*2*j*(2*j-1) mod n;
%p A205561     if member(x,S) then return j fi;
%p A205561     S:= S union {x}
%p A205561   od
%p A205561 end proc:
%p A205561 map(f, [$1..100]); # _Robert Israel_, Nov 18 2024
%t A205561 s = Table[(2n)!, {n, 1, 120}];
%t A205561 lk = Table[NestWhile[# + 1 &, 1,
%t A205561    Min[Table[Mod[s[[#]] - s[[j]], z], {j, 1, # - 1}]] =!= 0 &], {z, 1, Length[s]}]
%t A205561 Table[NestWhile[# + 1 &, 1,
%t A205561   Mod[s[[lk[[j]]]] - s[[#]], j] =!= 0 &],
%t A205561 {j, 1, Length[lk]}]
%t A205561 (* _Peter J. C. Moses_, Jan 27 2012 *)
%Y A205561 Cf. A002034, A010050, A204892, A205546, A378188, A378189.
%K A205561 nonn
%O A205561 1,1
%A A205561 _Clark Kimberling_, Feb 01 2012