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A212737
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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.
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12
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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
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OFFSET
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1,3
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LINKS
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FORMULA
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Formulae for the column sequences are given in A059912, A212906, ... .
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EXAMPLE
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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, ...
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MAPLE
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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);
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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