A207191 -id:A207191 - 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 *)

A207192 Numbers that match odd polynomials among the monic polynomials over {-1,0,1}, ordered as at A206821.

Original entry on oeis.org

2, 11, 13, 20, 57, 59, 65, 67, 90, 96, 98, 247, 249, 255, 257, 279, 281, 287, 289, 376, 382, 384, 406, 408, 414, 416, 1013, 1015, 1021, 1023, 1045, 1047, 1053, 1055, 1141, 1143, 1149, 1151, 1173, 1175, 1181, 1183, 1526, 1532, 1534, 1556, 1558

Offset: 1

Clark Kimberling, Feb 16 2012

nonn

The polynomials y(k,x) range through all monic polynomials with coefficients in {-1,0,1}, ordered as at A206821.

			The first 13 polynomials:
1 .... 1
2 .... x
3 .... x + 1
4 .... x^2
5 .... x^2 - 1
6 .... x^2 - x
7 .... x^2 - x - 1
8 .... x^2 + 1
9 .... x^2 + x
10 ... x^2 + x + 1
11 ... x^3
12 ... x^3 - 1
13 ... x^3 - x
Numbers n for which y(n,-x)=y(n,x): 1,4,5,8,26,...
Numbers n for which y(n,-x)=-y(n,x): 2,11,13,20,...

Cf. A206821.

t = Table[IntegerDigits[n, 2], {n, 1, 2000}];
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, 10}]
y = Flatten[y1]; (* polynomials over {-1,0,1} *)
Flatten[Position[y - (y /. x -> -x), 0]]  (* A207191 *)
Flatten[Position[y + (y /. x -> -x), 0]]  (* A207192  *)

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

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