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A206821 Numbers that match irreducible polynomials over {-1,0,1} with leading coefficient 1. 9
2, 3, 7, 8, 10, 14, 16, 18, 21, 23, 29, 31, 35, 41, 42, 44, 48, 50, 54, 56, 60, 62, 66, 70, 72, 76, 78, 80, 82, 84, 86, 88, 93, 97, 99, 103, 109, 111, 115, 117, 123, 125, 129, 131, 137, 141, 143, 147, 153, 155, 159, 161, 165, 167, 171, 173, 179, 183, 186, 188 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The monic polynomials y(n,x) having coefficients in {-1,0,1} are matched to the set N of positive integers as follows.  First, the monic polynomials p(n,x) having coefficients in {0,1} are matched to N as in A206074; i.e., the polynomial x^d(0) + x^d(1) + ... + d(n), where d(i) is 0 or 1 for 0<=i<=n and d(0)=1, matches the binary number d(0)d(1)...d(n).  Then monic polynomials having at least one negative coefficient are then inserted among the polynomials p(n,x), as follows:  x-1 goes between x and x+1, and for k>1, the polynomials x^k-p(n,x), for 0<n<2^k, go between x^k and x^k+1, in this order: x^k-p(1,x), x^k-p(2,x),..., x^k-p(2^k-1,x).  A program in the Mathematica section generates the resulting polynomials in the order just described.  The n-th polynomial, denoted here as y(n,x), can be obtained as y[[n]] from the program.  The first 11 polynomials, marked "yes" if irreducible over the field of rational numbers, are shown here:

n ..... y(n,x) ... irreducible

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

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

3 ..... 1+x ...... yes

4 ..... x^2 ...... no

5 .... -1+x^2 .... no

6 .... -x+x^2 .... no

7 .... -1-x+x^2 .. yes

8 ..... 1+x^2 .... yes

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

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

11 .... x^3 ...... no

...

Guide to sequences based on the polynomials y(n,x):

A206822, irreducible

A206829, number of distinct factors

A207187, multiples of x+1

A207188, multiples of x

A207189, multiples of x-1

A207190, multiples of x^2+1

A207191, even:  y(n,-x)=y(n,x)

A207192, odd: y(n,-x)=-y(n,x)

LINKS

Table of n, a(n) for n=1..60.

MATHEMATICA

t = Table[IntegerDigits[n, 2], {n, 1, 1000}];

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

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

TableForm[Table[{n, p[n], Factor[p[n]]}, {n, 1, 6}]]

f[k_] := 2^k - k; g[k_] := 2^k - 2 + f[k - 1];

q1[n_] := p[2^(k - 1)] - p[n + 1 - f[k]];

q2[n_] := p[n - f[k] + 2];

y1 = Table[p[n], {n, 1, 4}];

Do[AppendTo[y1, Join[Table[q1[n], {n, f[k], g[k] - 1}],

   Table[q2[n], {n, g[k], f[k + 1] - 1}]]], {k, 3, 8}]

y = Flatten[y1]; (* polynomials over {-1, 0, 1} *)

w = {}; Do[n++; If[IrreduciblePolynomialQ[y[[n]]], AppendTo[w, n]], {n, 200}]

w                          (* A206821 *)

Complement[Range[200], w]  (* A206822 *)

CROSSREFS

Cf. A206073, A206284, A206822.

Sequence in context: A225378 A028808 A029718 * A323003 A051468 A002274

Adjacent sequences:  A206818 A206819 A206820 * A206822 A206823 A206824

KEYWORD

nonn

AUTHOR

Clark Kimberling, Feb 12 2012

STATUS

approved

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Last modified August 18 15:00 EDT 2019. Contains 326106 sequences. (Running on oeis4.)