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%I #13 Aug 19 2021 10:56:54
%S 3,4,5,6,7,8,10,11,13,14,17,19,20,22,23,25,26,29,31,34,35,37,38,41,43,
%T 46,47,49,53,55,58,59,61,62,65,67,71,73,74,77,79,82,83,85,86,89,92,94,
%U 95,97,98,101,103,106,107,109,110,113,115,118,121,122,125,127
%N Numbers that match irreducible polynomials over {0,1,2}.
%C Each n > 1 matches a polynomial having coefficients in {0,1,2}, determined by the prime factorization of n.
%C Write n = p(1)^e(1) * p(2)^e(2) * ... * p(k)^e(k).
%C The matching polynomial is then
%C p(n,x) = e(1) + e(2)x + e(3)x^2 + ... + e(k)x^k.
%C Identities:
%C p(m*n) = p(m,x) + p(n,x);
%C p(m*n) = p(gcd(m,n),x) + p(lcm(m,n),x).
%C For an analogous enumeration of polynomials over {0,1}, see A206284.
%C "Irreducible" refers to irreducibility over the field of rational numbers.
%e Polynomials having coefficients in {0,1,2} are
%e matched to the positive integers as follows:
%e n ... p[n,x] .. irreducible
%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 .. 1+x^2 ... yes
%t t = Table[IntegerDigits[n, 3], {n, 1, 850}];
%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++; If[IrreduciblePolynomialQ[p[n, x]],
%t AppendTo[u, n]], {n, 300}]; u (* A207966 *)
%t Complement[Range[200], u] (* A207967 *)
%t b[n_] := FromDigits[IntegerDigits[u, 3][[n]]]
%t Table[b[n], {n, 1, 50}] (* A207968 *)
%Y Cf. A207967, A207968, A206284.
%K nonn
%O 1,1
%A _Clark Kimberling_, Feb 21 2012
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