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

1, 4, 5, 9, 14, 40, 20, 35, 44, 108, 65, 60, 45, 90, 119, 135, 60, 30, 189, 209, 46, 100, 63, 299, 145, 700, 100, 135, 390, 928, 99, 84, 522, 280, 629, 605, 56, 1480, 779, 740, 430, 684, 60, 989, 517, 80, 40, 1175, 195, 2548, 240, 252, 715, 424, 81, 1595, 220, 310

Offset: 1

Author

John H. Mason, Aug 09 2006

Keywords

• nonn

Comments

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.

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.

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).

Examples

			a(2)=4 because 2^4=16=1 mod 5 but 2^1, 2^2 and 2^3 are not;
a(3)=5 because 3^5=1 mod 11 and 5 is the smallest such.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000

Programs

Maple

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:
# second Maple program:
a:= n-> `if`(n=1, 1, numtheory[order](n, n^2+n-1)):
seq(a(n), n=1..75);  # Alois P. Heinz, Feb 18 2020

Mathematica

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 *)

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

Extensions

More terms from Klaus Brockhaus and Robert G. Wilson v, Aug 09 2006