A212737 Square array A(n,k), n>=1, k>=1, read by antidiagonals, where column k lists the orders of degree-d irreducible polynomials over GF(prime(k)); listing order for each column: ascending d, ascending value.

1, 1, 3, 1, 2, 7, 1, 2, 4, 5, 1, 2, 4, 8, 15, 1, 2, 3, 3, 13, 31, 1, 2, 5, 6, 6, 26, 9, 1, 2, 3, 10, 4, 8, 5, 21, 1, 2, 4, 4, 3, 8, 12, 10, 63, 1, 2, 3, 8, 6, 4, 12, 24, 16, 127, 1, 2, 11, 6, 16, 12, 6, 16, 31, 20, 17, 1, 2, 4, 22, 9, 3, 7, 8, 24, 62, 40, 51

Offset

1,3

Links

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Eric Weisstein's World of Mathematics, Irreducible Polynomial

Eric Weisstein's World of Mathematics, Polynomial Order

Formula

Formulae for the column sequences are given in A059912, A212906, ... .

Example

For k=1 the irreducible polynomials over GF(prime(1)) = GF(2) of degree 1-4 are: x, 1+x; 1+x+x^2; 1+x+x^3, 1+x^2+x^3; 1+x+x^2+x^3+x^4, 1+x+x^4, 1+x^3+x^4. The orders of these polynomials p (i.e., the smallest integer e for which p divides x^e+1) are 1; 3; 7; 5, 15. (Example: (1+x^3+x^4) * (1+x^3+x^4+x^6+x^8+x^9+x^10+x^11) == x^15+1 (mod 2)). Thus column k=1 begins: 1, 3, 7, 5, 15, ... .
Square array A(n,k) begins:
    1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
    3,  2,  2,  2,  2,  2,  2,  2,  2,  2, ...
    7,  4,  4,  3,  5,  3,  4,  3, 11,  4, ...
    5,  8,  3,  6, 10,  4,  8,  6, 22,  7, ...
   15, 13,  6,  4,  3,  6, 16,  9,  3, 14, ...
   31, 26,  8,  8,  4, 12,  3, 18,  4, 28, ...
    9,  5, 12, 12,  6,  7,  6,  4,  6,  3, ...
   21, 10, 24, 16,  8,  8,  9,  5,  8,  5, ...
   63, 16, 31, 24, 12, 14, 12,  8, 12,  6, ...
  127, 20, 62, 48, 15, 21, 18, 10, 16,  8, ...

Maple

with(numtheory):
M:= proc(n, i) M(n, i):= divisors(ithprime(i)^n-1) minus U(n-1, i) end:
U:= proc(n, i) U(n, i):= `if`(n=0, {}, M(n, i) union U(n-1, i)) end:
b:= proc(n, i) b(n, i):= sort([M(n, i)[]])[] end:
A:= proc() local l; l:= proc() [] end;
      proc(n, k) local t;
        if nops(l(k))<n then l(k):= [];
          for t while nops(l(k))<n
            do l(k):= [l(k)[], b(t, k)] od
        fi; l(k)[n]
      end:
    end():
seq(seq(A(n, 1+d-n), n=1..d), d=1..15);

Mathematica

m[n_, i_] := Divisors[Prime[i]^n-1] ~Complement~ u[n-1, i]; u[n_, i_] := u[n, i] = If[n == 0, {}, m[n, i] ~Union~ u[n-1, i]]; b[n_, i_] := Sort[m[n, i]]; a = Module[{l}, l[_] = {}; Function[{n, k}, Module[{t}, If [Length[l[k]] < n, l[k] = {}; For[t = 1, Length[l[k]] < n, t++, l[k] = Join[l[k], b[t, k]]]]; l[k][[n]]]]]; Table[Table[a[n, 1+d-n], {n, 1, d}], {d, 1, 15}] // Flatten (* Jean-François Alcover, Dec 20 2013, translated from Maple *)

Crossrefs

Columns k=1-10 give: A059912, A212906, A212485, A212486, A218336, A218337, A218338, A218339, A218340, A218341.

Sequence in context: A024743 A024963 A283435 * A307078 A134348 A331789

Adjacent sequences: A212734 A212735 A212736 * A212738 A212739 A212740

Keyword

nonn,look,tabl,changed

Author

Alois P. Heinz, Jun 02 2012

Status

approved