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 A213224 Minimal order A(n,k) of degree-n irreducible polynomials over GF(prime(k)); square array A(n,k), n>=1, k>=1, read by antidiagonals. 11
 1, 1, 3, 1, 4, 7, 1, 3, 13, 5, 1, 4, 31, 5, 31, 1, 3, 9, 13, 11, 9, 1, 7, 7, 5, 11, 7, 127, 1, 3, 9, 16, 2801, 7, 1093, 17, 1, 4, 307, 5, 25, 36, 19531, 32, 73, 1, 3, 27, 5, 30941, 9, 29, 32, 757, 11, 1, 3, 7, 16, 88741, 63, 43, 64, 19, 44, 23 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Maximal order of degree-n irreducible polynomials over GF(prime(k)) is prime(k)^n-1 and thus A(n,k) < prime(k)^n. LINKS Alois P. Heinz, Antidiagonals n = 1..45, flattened Eric Weisstein's World of Mathematics, Irreducible Polynomial Eric Weisstein's World of Mathematics, Polynomial Order FORMULA A(n,k) = min(M(n,k)) with M(n,k) = {d : d|(prime(k)^n-1)} \ U(n-1,k) and U(n,k) = M(n,k) union U(n-1,k) for n>0, U(0,k) = {}. EXAMPLE A(4,1) = 5: The degree-4 irreducible polynomials p over GF(prime(1)) = GF(2) are 1+x+x^2+x^3+x^4, 1+x+x^4, 1+x^3+x^4. Their orders (i.e., the smallest integer e for which p divides x^e+1) are 5, 15, 15, and the minimal order is 5. (1+x+x^2+x^3+x^4) * (1+x) == x^5+1 (mod 2). Square array A(n,k) begins: 1,      1,     1,    1,   1,       1,        1,   1, ... 3,      4,     3,    4,   3,       7,        3,   4, ... 7,     13,    31,    9,   7,       9,      307,  27, ... 5,      5,    13,    5,  16,       5,        5,  16, ... 31,    11,    11, 2801,  25,   30941,    88741, 151, ... 9,      7,     7,   36,   9,      63,        7,   7, ... 127, 1093, 19531,   29,  43, 5229043, 25646167, 701, ... 17,    32,    32,   64,  32,      32,      128,  17, ... MAPLE with(numtheory): M:= proc(n, i) option remember;       divisors(ithprime(i)^n-1) minus U(n-1, i)     end: U:= proc(n, i) option remember;       `if`(n=0, {}, M(n, i) union U(n-1, i))     end: A:= (n, k)-> min(M(n, k)[]): seq(seq(A(n, d+1-n), n=1..d), d=1..14); MATHEMATICA M[n_, i_] := 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]]; A[n_, k_] := Min[M[n, k]]; Table[Table[A[n, d+1-n], {n, 1, d}], {d, 1, 14}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *) CROSSREFS Columns k=1-10 give: A212953, A218356, A218357, A218358, A218359, A218360, A218361, A218362, A218363, A218364. Columns k=1-10 are first columns of: A059912, A212906, A212485, A212486, A218336, A218337, A218338, A218339, A218340, A218341. Cf. A212737 (all orders). Sequence in context: A049918 A028861 A081521 * A210218 A086273 A054143 Adjacent sequences:  A213221 A213222 A213223 * A213225 A213226 A213227 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Jun 06 2012 STATUS approved

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Last modified January 22 13:41 EST 2020. Contains 331149 sequences. (Running on oeis4.)