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%I #38 Dec 15 2015 00:55:15
%S 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 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 12,6,16,31,20,17,1,2,4,22,9,3,7,8,24,62,40,51
%N 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 Alois P. Heinz, <a href="/A212737/b212737.txt">Antidiagonals n = 1..141, flattened</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IrreduciblePolynomial.html">Irreducible Polynomial</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PolynomialOrder.html">Polynomial Order</a>
%F Formulae for the column sequences are given in A059912, A212906, ... .
%e 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 Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...
%e 7, 4, 4, 3, 5, 3, 4, 3, 11, 4, ...
%e 5, 8, 3, 6, 10, 4, 8, 6, 22, 7, ...
%e 15, 13, 6, 4, 3, 6, 16, 9, 3, 14, ...
%e 31, 26, 8, 8, 4, 12, 3, 18, 4, 28, ...
%e 9, 5, 12, 12, 6, 7, 6, 4, 6, 3, ...
%e 21, 10, 24, 16, 8, 8, 9, 5, 8, 5, ...
%e 63, 16, 31, 24, 12, 14, 12, 8, 12, 6, ...
%e 127, 20, 62, 48, 15, 21, 18, 10, 16, 8, ...
%p with(numtheory):
%p M:= proc(n, i) M(n, i):= divisors(ithprime(i)^n-1) minus U(n-1, i) end:
%p U:= proc(n, i) U(n, i):= `if`(n=0, {}, M(n, i) union U(n-1, i)) end:
%p b:= proc(n, i) b(n, i):= sort([M(n, i)[]])[] end:
%p A:= proc() local l; l:= proc() [] end;
%p proc(n, k) local t;
%p if nops(l(k))<n then l(k):= [];
%p for t while nops(l(k))<n
%p do l(k):= [l(k)[], b(t, k)] od
%p fi; l(k)[n]
%p end:
%p end():
%p seq(seq(A(n, 1+d-n), n=1..d), d=1..15);
%t 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 Columns k=1-10 give: A059912, A212906, A212485, A212486, A218336, A218337, A218338, A218339, A218340, A218341.
%K nonn,look,tabl
%O 1,3
%A _Alois P. Heinz_, Jun 02 2012
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