cp's OEIS Frontend

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.

%I A212737 #39 Feb 16 2025 08:33:17
%S A212737 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,
%T A212737 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,
%U A212737 12,6,16,31,20,17,1,2,4,22,9,3,7,8,24,62,40,51
%N 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.
%H A212737 Alois P. Heinz, <a href="/A212737/b212737.txt">Antidiagonals n = 1..141, flattened</a>
%H A212737 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IrreduciblePolynomial.html">Irreducible Polynomial</a>
%H A212737 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolynomialOrder.html">Polynomial Order</a>
%F A212737 Formulae for the column sequences are given in A059912, A212906, ... .
%e A212737 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, ... .
%e A212737 Square array A(n,k) begins:
%e A212737     1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
%e A212737     3,  2,  2,  2,  2,  2,  2,  2,  2,  2, ...
%e A212737     7,  4,  4,  3,  5,  3,  4,  3, 11,  4, ...
%e A212737     5,  8,  3,  6, 10,  4,  8,  6, 22,  7, ...
%e A212737    15, 13,  6,  4,  3,  6, 16,  9,  3, 14, ...
%e A212737    31, 26,  8,  8,  4, 12,  3, 18,  4, 28, ...
%e A212737     9,  5, 12, 12,  6,  7,  6,  4,  6,  3, ...
%e A212737    21, 10, 24, 16,  8,  8,  9,  5,  8,  5, ...
%e A212737    63, 16, 31, 24, 12, 14, 12,  8, 12,  6, ...
%e A212737   127, 20, 62, 48, 15, 21, 18, 10, 16,  8, ...
%p A212737 with(numtheory):
%p A212737 M:= proc(n, i) M(n, i):= divisors(ithprime(i)^n-1) minus U(n-1, i) end:
%p A212737 U:= proc(n, i) U(n, i):= `if`(n=0, {}, M(n, i) union U(n-1, i)) end:
%p A212737 b:= proc(n, i) b(n, i):= sort([M(n, i)[]])[] end:
%p A212737 A:= proc() local l; l:= proc() [] end;
%p A212737       proc(n, k) local t;
%p A212737         if nops(l(k))<n then l(k):= [];
%p A212737           for t while nops(l(k))<n
%p A212737             do l(k):= [l(k)[], b(t, k)] od
%p A212737         fi; l(k)[n]
%p A212737       end:
%p A212737     end():
%p A212737 seq(seq(A(n, 1+d-n), n=1..d), d=1..15);
%t A212737 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 *)
%Y A212737 Columns k=1-10 give: A059912, A212906, A212485, A212486, A218336, A218337, A218338, A218339, A218340, A218341.
%K A212737 nonn,look,tabl
%O A212737 1,3
%A A212737 _Alois P. Heinz_, Jun 02 2012