%I A127699
%S A127699 1,2,6,4,20,6,42,8,18,20,220,12,156,42,60,16,272,18,342,20,42,220,5060,
%T A127699 24,100,156,54,84,2436,60,1860,32,660,272,420,36,1332,342,156,40,1640,
%U A127699 42,1806,220,180,5060,237820,48,294,100,816,156,8268,54,220,168
%N A127699 Length of period of the sequence (1^1^1^..., 2^2^2^..., 3^3^3^..., 4^4^4^..., 
               ...) modulo n.
%C A127699 For any positive integers a and m the sequence a, a^a, a^a^a, a^a^a^a,
               ... becomes eventually constant modulo m. So the remainder of a^a^a^... 
               modulo n is well-defined.
%C A127699 Shapiro and Shapiro treat this problem. [From T. D. Noe (noe(AT)sspectra.com), 
               Jan 30 2009]
%H A127699 T. D. Noe, <a href="b127699.txt">Table of n, a(n) for n=1..1000</a>
%H A127699 Daniel B. Shapiro and S. David Shapiro, <a href="http://www.emis.de/journals/
               INTEGERS/vol7.html">Iterated Exponents in Number Theory,</a> Integers 
               7 (2007), #A23. [From T. D. Noe (noe(AT)sspectra.com), Jan 30 2009]
%F A127699 a(n) = lcm(n, a(lambda(n))), where lambda is Carmichael's reduced totient 
               function. [From T. D. Noe (noe(AT)sspectra.com), Jan 30 2009]
%e A127699 a(10)=20 because the last digit of 1^1^1^.. is 1; the sequence 2,2^2,
               2^2^2,.. ends with 2,4,6,6,...; the sequence 3,3^3,3^3^3,... with 
               3,7,7,...; 4,4^4,4^4^4,... with 4,6,6,...; and so on. We get as last 
               digits 1,6,7,6,5,6,3,6,9,0, 1,6,3,6,5,6,7,6,9,0 and then the pattern 
               repeats.
%t A127699 nn=100; a=Table[0,{nn}]; a[[1]]=1; Do[a[[n]]=LCM[n,a[[CarmichaelLambda[n]]]], 
               {n,2,nn}]; a [From T. D. Noe (noe(AT)sspectra.com), Jan 30 2009]
%Y A127699 Cf. A000010.
%Y A127699 Sequence in context: A100695 A100140 A009262 this_sequence A124838 A088659 
               A052100
%Y A127699 Adjacent sequences: A127696 A127697 A127698 this_sequence A127700 A127701 
               A127702
%K A127699 easy,nonn
%O A127699 1,2
%A A127699 Jan Fricke (jfricke(AT)math.uni-jena.de), Apr 11 2007
%E A127699 Extension and correction from T. D. Noe (noe(AT)sspectra.com), Jan 30 
               2009
%E A127699 Removed incorrect formula -- T. D. Noe (noe(AT)sspectra.com), Feb 02 
               2009