A207669 - OEIS

%I #15 Aug 19 2021 10:56:25

%S 3,4,5,6,7,8,10,14,17,20,22,25,34,35,38,41,43,46,49,53,58,59,65,67,71,

%T 73,77,79,86,89,92,94,97,101,110,115,118,121,125,134,137,139,145,149,

%U 151,158,166,169,172,181,185,188,190,197,205,209,212,214,217

%N Numbers that match polynomials irreducible (mod 3), with coefficients in {0,1,2}.

%C For such polynomials irreducible over the field of rational numbers, see A207966, which also describes the enumeration of all the nonzero polynomials whose coefficients are all in {0,1,2}.

%e Polynomials having coefficients in {0,1,2} are

%e enumerated by the positive integers as follows:

%e n ... p[n,x] .. irreducible (mod 3)

%e 1 ... 1 ....... no

%e 2 ... 2 ....... no

%e 3 ... x ....... yes

%e 4 ... x+1 ..... yes

%e 5 ... x+2 ..... yes

%e 6 ... 2x ...... yes

%e 7 ... 2x+1 .... yes

%e 8 ... 2x+2 .... yes

%e 9 ... x^2 ..... no

%e 10 .. x^2+1 ... yes

%e 11 .. x^2+2 ... no

%e The least n for which p(n,x) is irreducible over the

%e rationals but not modulo 3 is 13; the factorization of

%e p(13,x) is (x+1)(x+2) (mod 3).

%t t = Table[IntegerDigits[n, 3], {n, 1, 1000}];

%t b[n_] := Reverse[Table[x^k, {k, 0, n}]]

%t p[n_, x_] := t[[n]].b[-1 + Length[t[[n]]]]

%t Table[p[n, x], {n, 1, 15}]

%t u = {}; Do[n++;

%t If[IrreduciblePolynomialQ[p[n, x], Modulus -> 3],

%t   AppendTo[u, n]], {n, 1, 400}]

%t u                           (* A207669 *)

%t Complement[Range[200], %]   (* A207670 *)

%t b[n_] := FromDigits[IntegerDigits[u, 3][[n]]]

%t Table[b[n], {n, 1, 50}]     (* A207671 *)

%Y Cf. A207670 (complement), A207671 (ternary).

%K nonn

%O 1,1

%A _Clark Kimberling_, Feb 26 2012