A206822 -id:A206822 - cp's OEIS frontend

A206821 Numbers that match irreducible polynomials over {-1,0,1} with leading coefficient 1.

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

Offset: 1

Clark Kimberling, Feb 12 2012

nonn

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 ..... 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)

Cf. A206073, A206284, A206822.

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 *)

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A206821 Numbers that match irreducible polynomials over {-1,0,1} with leading coefficient 1.

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