

A212953


Minimal order of degreen irreducible polynomials over GF(2).


3



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
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OFFSET

1,2


COMMENTS

a(n) = smallest odd m such that A002326((m1)/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^n1 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

Alois P. Heinz, Table of n, a(n) for n = 1..179
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^n1)} \ U(n1) and U(n) = M(n) union U(n1) for n>0, U(0) = {}.
a(n) = A059912(n,1) = A213224(n,1).


EXAMPLE

For n=4 the degree4 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^n1) minus U(n1)
end:
U:= proc(n) option remember;
`if`(n=0, {}, M(n) union U(n1))
end:
a:= n> min(M(n)[]):
seq(a(n), n=1..50);


MATHEMATICA

M[n_] := M[n] = Divisors[2^n1] ~Complement~ U[n1];
U[n_] := U[n] = If[n == 0, {}, M[n] ~Union~ U[n1]];
a[n_] := Min[M[n]];
Array[a, 50] (* JeanFrançois Alcover, Mar 22 2017, translated from Maple *)


CROSSREFS

Cf. 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



