

A014580


Binary irreducible polynomials (primes in the ring GF(2)[X]), evaluated at X=2.


103



2, 3, 7, 11, 13, 19, 25, 31, 37, 41, 47, 55, 59, 61, 67, 73, 87, 91, 97, 103, 109, 115, 117, 131, 137, 143, 145, 157, 167, 171, 185, 191, 193, 203, 211, 213, 229, 239, 241, 247, 253, 283, 285, 299, 301, 313, 319, 333, 351, 355, 357, 361, 369, 375
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OFFSET

1,1


COMMENTS

Or, binary irreducible polynomials, interpreted as binary vectors, then written in base 10.
2^n  1 is a term if and only if n = 2 or n is a prime and 2 is a primitive root modulo n.  Jianing Song, May 10 2021


LINKS



EXAMPLE

x^4 + x^3 + 1 > 16+8+1 = 25. Or, x^4 + x^3 + 1 > 11001 (binary) = 25 (decimal).


MATHEMATICA

fQ[n_] := Block[{ply = Plus @@ (Reverse@ IntegerDigits[n, 2] x^Range[0, Floor@ Log2@ n])}, ply == Factor[ply, Modulus > 2] && n != 2^Floor@ Log2@ n]; fQ[2] = True; Select[ Range@ 378, fQ] (* Robert G. Wilson v, Aug 12 2011 *)
Reap[Do[If[IrreduciblePolynomialQ[IntegerDigits[n, 2] . x^Reverse[Range[0, Floor[Log[2, n]]]], Modulus > 2], Sow[n]], {n, 2, 1000}]][[2, 1]] (* JeanFrançois Alcover, Nov 21 2016 *)


PROG



CROSSREFS

Number of degreen irreducible polynomials: A001037, see also A000031.
Table of irreducible factors of n: A256170.
Sequences analyzing the difference between factorization into GF(2)[X] irreducibles and ordinary prime factorization of the corresponding integer: A234741, A234742, A235032, A235033, A235034, A235035, A235040, A236850, A325386, A325559, A325560, A325563, A325641, A325642, A325643.
See A115871 for sequences related to crossdomain congruences.


KEYWORD

nonn


AUTHOR

David Petry (petry(AT)accessone.com)


STATUS

approved



