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A205546 Least positive integer k such that n divides k^k-j^j for some j in [1,k-1].

%I A205546 #14 Aug 23 2023 16:28:26
%S A205546 2,3,2,4,4,4,4,6,4,6,5,4,3,4,4,6,4,4,5,6,4,5,3,8,6,3,6,4,6,8,6,6,6,8,
%T A205546 6,4,7,9,8,6,9,4,6,5,8,8,10,8,8,6,4,9,9,9,8,8,8,6,9,8,10,12,4,6,8,9,9,
%U A205546 8,8,8,5,10,10,9,12,9,8,9,9,6,9,9,18,4,4,16,7,8,8,12,8,8,12,10
%N A205546 Least positive integer k such that n divides k^k-j^j for some j in [1,k-1].
%C A205546 For a guide to related sequences, see A204892.
%H A205546 Robert Israel, <a href="/A205546/b205546.txt">Table of n, a(n) for n = 1..10000</a>
%e A205546 1 divides 2^2-1^1 -> k=2, j=1
%e A205546 2 divides 3^3-1^1 -> k=3, j=1
%e A205546 3 divides 2^2-1^1 -> k=2, j=1
%e A205546 4 divides 4^4-2^2 -> k=4, j=2
%p A205546 f:= proc(n) local S,k,v;
%p A205546   S:= {}:
%p A205546   for k from 1 do
%p A205546     v:= k &^ k mod n;
%p A205546     if member(v,S) then return k fi;
%p A205546     S:= S union {v}
%p A205546   od
%p A205546 end proc:
%p A205546 map(f, [$1..100]); # _Robert Israel_, Aug 23 2023
%t A205546 s = Table[n^n, {n, 1, 120}];
%t A205546 lk = Table[NestWhile[# + 1 &, 1,
%t A205546    Min[Table[Mod[s[[#]] - s[[j]], z], {j, 1, # - 1}]] =!= 0 &], {z, 1, Length[s]}]
%t A205546 Table[NestWhile[# + 1 &, 1,
%t A205546   Mod[s[[lk[[j]]]] - s[[#]], j] =!= 0 &],
%t A205546 {j, 1, Length[lk]}]
%t A205546 (* _Peter J. C. Moses_, Jan 27 2012 *)
%Y A205546 Cf. A204892, A000312.
%K A205546 nonn
%O A205546 1,1
%A A205546 _Clark Kimberling_, Jan 31 2012