%I A118106
%S A118106 1,1,1,1,1,2,1,1,1,4,1,2,1,3,4,1,1,6,1,4,6,10,1,2,1,12,1,6,1,4,1,1,10,
               8,
%T A118106 12,6,1,18,3,4,1,6,1,10,12,11,1,4,1,20,16,12,1,18,5,6,18,28,1,4,1,5,6,
               1,
%U A118106 4,10,1,8,22,12,1,6,1,36,20,18,30,12,1,4,1,20,1,6,16,14,28,10,1,12,12
%N A118106 Period of the vector sequence d(n)^k mod n for k=1,2,3,..., where d(n) 
               is the vector of divisors of n.
%C A118106 This sequence is related to the period of sigma_k(n) mod n. Note that 
               a(n)=1 iff n is a power of a prime.
%C A118106 The record period lengths of p-1 occur at n=2p, where p is a prime with 
               primitive root 2 (A001122). - T. D. Noe, Oct 25 2007
%H A118106 T. D. Noe, <a href="b118106.txt">Table of n, a(n) for n=1..1000</a>
%e A118106 a(35)=12 because d(35)=(1,5,7,35) and (1,5,7,35)^k (mod 35) is the sequence 
               of vectors (1,5,7,0), (1,25,14,0), (1,20,28,0), (1,30,21,0), (1,10,
               7,0), (1,15,14,0), (1,5,28,0), (1,25,21,0), (1,20,7,0), (1,30,14,
               0), (1,10,28,0), (1,15,21,0), (1,5,7,0),..., which has a period of 
               12.
%t A118106 Table[d=Divisors[n]; k=0; found=False; While[i=0; While[i<k-1 && !found, 
               i++; found=(dk[i]==dk[k])]; !found, k++; dk[k]=PowerMod[d,k,n]]; 
               k-i, {n,100}]
%Y A118106 Cf. A118107 (period of the vector sequence d(n)^2^k mod n).
%Y A118106 Sequence in context: A156141 A157113 A139320 this_sequence A143201 A158298 
               A112331
%Y A118106 Adjacent sequences: A118103 A118104 A118105 this_sequence A118107 A118108 
               A118109
%K A118106 nonn
%O A118106 1,6
%A A118106 T. D. Noe (noe(AT)sspectra.com), Apr 13 2006