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A059912
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Triangle T(n,k) of orders of n degree irreducible polynomials over GF(2) listed in ascending order, k=1..A059499(n).
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9
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1, 3, 7, 5, 15, 31, 9, 21, 63, 127, 17, 51, 85, 255, 73, 511, 11, 33, 93, 341, 1023, 23, 89, 2047, 13, 35, 39, 45, 65, 91, 105, 117, 195, 273, 315, 455, 585, 819, 1365, 4095, 8191, 43, 129, 381, 5461, 16383, 151, 217, 1057, 4681, 32767, 257, 771, 1285, 3855
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OFFSET
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1,2
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LINKS
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Alois P. Heinz, Rows n = 1..71, flattened
Polynomial order: computes the order of an irreducible polynomial over a finite field GF(p).
Eric Weisstein's World of Mathematics, Irreducible Polynomial
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FORMULA
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T(n,k) = k-th smallest element of M(n) = {d : d|(2^n-1)} \ U(n-1) with U(n) = M(n) union U(n-1) if n>0, U(0) = {}. - Alois P. Heinz, Jun 01 2012
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EXAMPLE
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Triangle begins:
1;
3;
7;
5, 15;
31;
9, 21, 63;
127;
17, 51, 85, 255;
73, 511;
11, 33, 93, 341, 1023;
There are 18 (cf. A001037) irreducible polynomials of degree 7 over GF(2) which all have order 127.
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MAPLE
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with (numtheory):
M:= proc(n) option remember;
divisors(2^n-1) minus U(n-1)
end:
U:= proc(n) option remember;
`if`(n=0, {}, M(n) union U(n-1))
end:
T:= n-> sort([M(n)[]])[]:
seq (T(n), n=1..20); # Alois P. Heinz, May 31 2012
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MATHEMATICA
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m[n_] := m[n] = Complement[ Divisors[2^n - 1], u[n - 1]]; u[0] = {}; u[n_] := u[n] = Union[ m[n], u[n - 1]]; t[n_, k_] := m[n][[k]]; Flatten[ Table[t[n, k], {n, 1, 16}, {k, 1, Length[ m[n] ]}]] (* Jean-François Alcover, Jun 14 2012, after Alois P. Heinz *)
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CROSSREFS
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Cf. A058943, A059478, A059499, A001037, A059913.
Cf. A212906, A212485, A212486.
Column k=1 of A212737.
Column k=1 gives: A212953.
Last elements of rows give: A000225.
Sequence in context: A090940 A090916 A184162 * A115765 A112071 A046561
Adjacent sequences: A059909 A059910 A059911 * A059913 A059914 A059915
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KEYWORD
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easy,nonn,tabf,changed
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AUTHOR
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Vladeta Jovovic, Feb 09 2001
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STATUS
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approved
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