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A207966 Numbers that match irreducible polynomials over {0,1,2}. 5
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, 46, 47, 49, 53, 55, 58, 59, 61, 62, 65, 67, 71, 73, 74, 77, 79, 82, 83, 85, 86, 89, 92, 94, 95, 97, 98, 101, 103, 106, 107, 109, 110, 113, 115, 118, 121, 122, 125, 127 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Each n > 1 matches a polynomial having coefficients in {0,1,2}, determined by the prime factorization of n.

Write n = p(1)^e(1) * p(2)^e(2) * ... * p(k)^e(k).

The matching polynomial is then

p(n,x) = e(1) + e(2)x + e(3)x^2 + ... + e(k)x^k.

Identities:

p(m*n) = p(m,x) + p(n,x);

p(m*n) = p(gcd(m,n),x) + p(lcm(m,n),x).

For an analogous enumeration of polynomials over {0,1}, see A206284.

"Irreducible" refers to irreducibility over the field of rational numbers.

LINKS

Table of n, a(n) for n=1..64.

EXAMPLE

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

matched to the positive integers as follows:

n ... p[n,x] .. irreducible

1 ... 1 ....... no

2 ... 2 ....... no

3 ... x ....... yes

4 ... x+1 ..... yes

5 ... x+2 ..... yes

6 ... 2x ...... yes

7 ... 2x+1 .... yes

8 ... 2x+2 .... yes

9 ... x^2 ..... no

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

MATHEMATICA

t = Table[IntegerDigits[n, 3], {n, 1, 850}];

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

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

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

u = {}; Do[n++; If[IrreduciblePolynomialQ[p[n, x]],

  AppendTo[u, n]], {n, 300}]; u  (* A207966 *)

Complement[Range[200], u]        (* A207967 *)

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

Table[b[n], {n, 1, 50}]          (* A207968 *)

CROSSREFS

Cf. A207967, A207968, A206284.

Sequence in context: A073071 A039134 A298003 * A097901 A026423 A026427

Adjacent sequences:  A207963 A207964 A207965 * A207967 A207968 A207969

KEYWORD

nonn

AUTHOR

Clark Kimberling, Feb 21 2012

STATUS

approved

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Last modified December 6 16:21 EST 2021. Contains 349567 sequences. (Running on oeis4.)