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
Examples
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, ...
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
Crossrefs
Programs
-
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)) -
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 *)