A205783 Complement of A206074, a coding of reducible polynomials over Q (with coefficients 0 or 1).

1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 70, 72, 74, 75, 76, 78, 80, 82, 84, 85, 86, 88, 90, 92, 93, 94, 95, 96, 98, 99, 100

Offset: 1

Author

Clark Kimberling, Feb 03 2012

Keywords

• nonn

Comments

Reducibility here refers to the field of rational numbers.

Except for its initial 3, is A039004 a subsequence of A205783?

Examples

			The reducible polynomials matching the first four terms:
1 = 1(base 2) matches 1
4 = 100(base 2) matches x^2
6 = 110(base 2) matches x^2 + x
8 = 1000(base 2) matches x^3
9 = 1001(base 2) matches x^3 + 1
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..21951

Crossrefs

Cf. A206074 (complement), A255573 (left inverse).

After 1 a subsequence of A091212 (69 is the first term missing from here).

Cf. also permutations A260421 - A260426.

Programs

Mathematica

t = Table[IntegerDigits[n, 2], {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                          (* A206074 *)
Complement[Range[200], u]  (* A205783 *)
b[n_] := FromDigits[IntegerDigits[u, 2][[n]]]
Table[b[n], {n, 1, 40}]    (* A206073 *)

PARI

isA205783(n) = ((n > 0) && !polisirreducible(Pol(binary(n))));
n = 0; i = 0; while(n < 32768, n++; if(isA205783(n), i++; write("b205783.txt", i, " ", n)));
\\ Antti Karttunen, Jul 28 2015 after Joerg Arndt's code for A206074.

Formula

Other identities and observations. For all n >= 1:

A255573(a(n)) = n.