%I A121052
%S A121052 1,4,5,9,14,40,20,35,44,108,65,60,45,90,119,135,60,30,189,209,46,100,63,
%T A121052 299,145,700,100,135,390,928,99,84,522,280,629,605,56,1480,779,740,430,
%U A121052 684,60,989,517,80,40,1175,195,2548,240,252,715,424,81,1595,220,310
%N A121052 Smallest positive integer m for which n^m is congruent to 1 modulo n^2+n-1.
%C A121052 The sequence arises as the order of a shuffle of n(n+1) cards in which 
               cards are laid out in an array of n+1 rows of n columns; cards are 
               picked up by column and laid out by rows.
%C A121052 More generally there is a function of two variables, f(r,c) for which 
               f(r,c) is the least integer such that c^f(r,c) is congurent to 1 
               modulo rc-1. Of interest is the ratio of phi(rc-1)/f(r,c) or in the 
               case of the sequence proposed, phi(n^2+n-1)/m.
%C A121052 I would like to know if there is some direct way to predict these orders, 
               or the ratio of phi(rc-1)/f(r,c). The program provided produces the 
               table f(r,c).
%e A121052 a(2)=4 because 2^4=16=1 mod 5 but 2^1, 2^2 and 2^3 are not;
%e A121052 a(3)=5 because 3^5=1 mod 11 and 5 is the smallest such.
%p A121052 TAB:=proc(Rmin,Rmax,Cmin,Cmax) local r,c,T,m,ct,A; T:=array(1..Rmax-Rmin+1,
               1..Cmax-Cmin+1); for r from Rmin to Rmax do for c from Cmin to Cmax 
               do A:=c;ct:=1;m:=r*c-1; while not A = 1 do A:=A*c mod m;ct:=ct+1; 
               od; T[r-Rmin+1,c-Cmin+1]:=[ct,phi(m)]; od;od; eval(T) end:
%t A121052 f[n_] := If[n == 1, 1, Block[{m = 1, k = n^2 + n - 1}, While[Mod[n^m, 
               k] != 1, m++ ]; m]]; Array[f, 59] (* Robert G. Wilson v *)
%o A121052 (PARI) print1(1,",");for(n=2,60,q=n^2+n-1;m=1;while(lift(Mod(n,q)^m)!=1,
               m++);print1(m,",")) - (Klaus Brockhaus, Aug 09 2006)
%Y A121052 Sequence in context: A041493 A042765 A041353 this_sequence A041823 A042489 
               A049860
%Y A121052 Adjacent sequences: A121049 A121050 A121051 this_sequence A121053 A121054 
               A121055
%K A121052 nonn
%O A121052 1,2
%A A121052 John Mason (j.h.mason(AT)open.ac.uk), Aug 09 2006
%E A121052 More terms from Klaus Brockhaus (klaus-brockhaus(AT)t-online.de) and 
               Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 09 2006