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A121052 Smallest positive integer m for which n^m is congruent to 1 modulo n^2+n-1.

%I A121052 #16 Apr 30 2025 14:30:53
%S A121052 1,4,5,9,14,40,20,35,44,108,65,60,45,90,119,135,60,30,189,209,46,100,
%T A121052 63,299,145,700,100,135,390,928,99,84,522,280,629,605,56,1480,779,740,
%U A121052 430,684,60,989,517,80,40,1175,195,2548,240,252,715,424,81,1595,220,310
%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 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 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).
%H A121052 Alois P. Heinz, <a href="/A121052/b121052.txt">Table of n, a(n) for n = 1..20000</a>
%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:
%p A121052 # second Maple program:
%p A121052 a:= n-> `if`(n=1, 1, numtheory[order](n, n^2+n-1)):
%p A121052 seq(a(n), n=1..75);  # _Alois P. Heinz_, Feb 18 2020
%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
%K A121052 nonn
%O A121052 1,2
%A A121052 _John H. Mason_, Aug 09 2006
%E A121052 More terms from _Klaus Brockhaus_ and _Robert G. Wilson v_, Aug 09 2006