OFFSET
1,24
COMMENTS
Fix n. Start with k (0 <= k <= n-1) and repeatedly square and reduce mod n until this repeats; T(n,k) is the length of the cycle that is reached.
A279186 gives maximal entry in each row.
A037178 gives maximal entry in row p, p = n-th prime.
A279187 gives maximal entry in row c, c = n-th composite number.
A279188 gives maximal entry in row c, c = prime(n)^2.
A256608 gives LCM of entries in row n.
A256607 gives T(2,n).
LINKS
Robert Israel, Table of n, a(n) for n = 1..10011 (rows 1 to 141, flattened)
Haifeng Xu, The largest cycles consist by the quadratic residues and Fermat primes, arXiv:1601.06509 [math.NT], 2016.
EXAMPLE
The triangle begins:
1,
1,1,
1,1,1,
1,1,1,1,
1,1,1,1,1,
1,1,1,1,1,1,
1,1,2,2,2,2,1,
1,1,1,1,1,1,1,1,
1,1,2,1,2,2,1,2,1,
1,1,1,1,1,1,1,1,1,1,
1,1,4,4,4,4,4,4,4,4,1,
...
For example, if n=11 and k=2, repeatedly squaring mod 11 gives the sequence 2, 4, 5, 3, 9, 4, 5, 3, 9, 4, 5, 3, ..., which has period T(11,2) = 4.
MAPLE
A279185 := proc(k, n) local g, y, r;
if k = 0 then return 1 fi;
y:= n;
g:= igcd(k, y);
while g > 1 do
y:= y/g;
g:= igcd(k, y);
od;
if y = 1 then return 1 fi;
r:= numtheory:-order(k, y);
r:= r/2^padic:-ordp(r, 2);
if r = 1 then return 1 fi;
numtheory:-order(2, r)
end proc:
seq(seq(A279185(k, n), k=0..n-1), n=1..20); # Robert Israel, Dec 14 2016
MATHEMATICA
T[n_, k_] := Module[{g, y, r}, If[k == 0, Return[1]]; y = n; g = GCD[k, y]; While[g > 1, y = y/g; g = GCD[k, y]]; If[y == 1, Return[1]]; r = MultiplicativeOrder[k, y]; r = r/2^IntegerExponent[r, 2]; If[r == 1, Return[1]]; MultiplicativeOrder[2, r]];
Table[T[n, k], {n, 1, 13}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Nov 27 2017, after Robert Israel *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Dec 14 2016
STATUS
approved