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 A212953 Minimal order of degree-n irreducible polynomials over GF(2). 6
 1, 3, 7, 5, 31, 9, 127, 17, 73, 11, 23, 13, 8191, 43, 151, 257, 131071, 19, 524287, 25, 49, 69, 47, 119, 601, 2731, 262657, 29, 233, 77, 2147483647, 65537, 161, 43691, 71, 37, 223, 174763, 79, 187, 13367, 147, 431, 115, 631, 141, 2351, 97, 4432676798593, 251 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS a(n) = smallest odd m such that A002326((m-1)/2) = n. - Thomas Ordowski, Feb 04 2014 For n > 1; n < a(n) < 2^n, wherein a(n) = n+1 iff n+1 is A001122 a prime with primitive root 2, or a(n) = 2^n-1 iff n is a Mersenne exponent A000043. - Thomas Ordowski, Feb 08 2014 REFERENCES W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, Springer 2004, Third Edition, 4.3 Factorization of Prime Ideals in Extensions. More About the Class Group (Theorem 4.33), 4.4 Notes to Chapter 4 (Theorem 4.40). - Regarding the first comment. LINKS Max Alekseyev, Table of n, a(n) for n = 1..1236 (first 179 terms from Alois P. Heinz) Eric Weisstein's World of Mathematics, Irreducible Polynomial Eric Weisstein's World of Mathematics, Polynomial Order FORMULA a(n) = min(M(n)) with M(n) = {d : d|(2^n-1)} \ U(n-1) and U(n) = M(n) union U(n-1) for n>0, U(0) = {}. a(n) = A059912(n,1) = A213224(n,1). EXAMPLE For n=4 the degree-4 irreducible polynomials p over 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. (Example: (1+x+x^2+x^3+x^4) * (1+x) == x^5+1 (mod 2)). Thus the minimal order is 5 and a(4) = 5. MAPLE 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: a:= n-> min(M(n)[]): seq(a(n), n=1..50); MATHEMATICA M[n_] := M[n] = Divisors[2^n-1] ~Complement~ U[n-1]; U[n_] := U[n] = If[n == 0, {}, M[n] ~Union~ U[n-1]]; a[n_] := Min[M[n]]; Array[a, 50] (* Jean-François Alcover, Mar 22 2017, translated from Maple *) CROSSREFS Cf. A003060, A059912, A213224. Sequence in context: A186522 A048857 A005420 * A161818 A161509 A108974 Adjacent sequences: A212950 A212951 A212952 * A212954 A212955 A212956 KEYWORD nonn AUTHOR Alois P. Heinz, Jun 01 2012 STATUS approved

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Last modified February 28 07:03 EST 2024. Contains 370387 sequences. (Running on oeis4.)