%I #13 Feb 18 2020 19:32:57
%S 1,4,5,9,14,40,20,35,44,108,65,60,45,90,119,135,60,30,189,209,46,100,
%T 63,299,145,700,100,135,390,928,99,84,522,280,629,605,56,1480,779,740,
%U 430,684,60,989,517,80,40,1175,195,2548,240,252,715,424,81,1595,220,310
%N Smallest positive integer m for which n^m is congruent to 1 modulo n^2+n-1.
%C 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 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 congruent 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 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).
%H Alois P. Heinz, <a href="/A121052/b121052.txt">Table of n, a(n) for n = 1..20000</a>
%e a(2)=4 because 2^4=16=1 mod 5 but 2^1, 2^2 and 2^3 are not;
%e a(3)=5 because 3^5=1 mod 11 and 5 is the smallest such.
%p 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:
%p # second Maple program:
%p a:= n-> `if`(n=1, 1, numtheory[order](n, n^2+n-1)):
%p seq(a(n), n=1..75); # _Alois P. Heinz_, Feb 18 2020
%t 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 (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)
%K nonn
%O 1,2
%A _John H. Mason_, Aug 09 2006
%E More terms from _Klaus Brockhaus_ and _Robert G. Wilson v_, Aug 09 2006