OFFSET
1,2
COMMENTS
A permutation of the odd positive numbers; namely, order each odd number d by the multiplicative order of 2 modulo d (in case of a tie, smaller d go first). - Jeppe Stig Nielsen, Feb 13 2020
LINKS
Alois P. Heinz, Rows n = 1..71, flattened
Eric Weisstein's World of Mathematics, Irreducible Polynomial
FORMULA
T(n,k) = k-th smallest element of M(n) = {d : d|(2^n-1)} \ U(n-1) with U(n) = M(n) union U(n-1) if n>0, U(0) = {}. - Alois P. Heinz, Jun 01 2012
EXAMPLE
There are 18 (cf. A001037) irreducible polynomials of degree 7 over GF(2) which all have order 127.
Triangle T(n,k) begins:
1;
3;
7;
5, 15;
31;
9, 21, 63;
127;
17, 51, 85, 255;
73, 511;
11, 33, 93, 341, 1023;
...
MAPLE
with(numtheory):
M:= proc(n) option remember;
divisors(2^n-1) minus U(n-1)
end:
U:= proc(n) option remember;
`if`(n=0, {}, M(n) union U(n-1))
end:
T:= n-> sort([M(n)[]])[]:
seq(T(n), n=1..20); # Alois P. Heinz, May 31 2012
MATHEMATICA
m[n_] := m[n] = Complement[ Divisors[2^n - 1], u[n - 1]]; u[0] = {}; u[n_] := u[n] = Union[ m[n], u[n - 1]]; t[n_, k_] := m[n][[k]]; Flatten[ Table[t[n, k], {n, 1, 16}, {k, 1, Length[ m[n] ]}]] (* Jean-François Alcover, Jun 14 2012, after Alois P. Heinz *)
PROG
(PARI) maxDegree=26; for(n=1, maxDegree, forstep(d=1, 2^n, 2, znorder(Mod(2, d))==n&&print1(d, ", "))) \\ inefficient, Jeppe Stig Nielsen, Feb 13 2020
CROSSREFS
KEYWORD
AUTHOR
Vladeta Jovovic, Feb 09 2001
STATUS
approved