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 A207813 Numbers that match irreducible Zeckendorf polynomials. 7

%I

%S 2,4,9,17,19,25,27,30,40,43,46,53,56,59,61,67,69,72,77,82,85,93,95,98,

%T 101,103,108,111,114,119,124,129,135,137,140,150,153,161,166,169,171,

%U 177,179,182,187,195,197,205,208,211,213,218,224,229,237,239

%N Numbers that match irreducible Zeckendorf polynomials.

%C The Zeckendorf representation of a positive integer n is

%C a unique sum

%C ...

%C c(k-2)F(k) + c(k-3)F(k-1) + ... + c(1)F(3) + c(0)F(2),

%C ...

%C where F=A000045 (Fibonacci numbers), c(k-2)=1, and for

%C j=0,1,...,k-3, there are two restrictions on

%C coefficients: c(j) is 0 or 1, and c(j)c(j+1)=0; viz.,

%C no two consecutive Fibonacci numbers appear. The

%C Zeckendorf polynomial Z(n,x) is introduced here as

%C ...

%C c(k-2)x^(k-2) + c(k-3)x^(k-3) + ... + c(1)x + c(0) .

%C ...

%C A207813 refers to irreduciblity over the field of

%C rational numbers.

%e n...k...Z(n)....Z(n,x).......irreducible

%e 1...2...1.......1............no

%e 2...3...10......x............yes

%e 3...4...100.....x^2..........no

%e 4...4...101.....x^2 + 1......yes

%e 5...5...1000....x^3..........no

%e 6...5...1001....x^3 + 1......no

%e 7...5...1010....x^3 + x .....no

%e 8...5...10000...x^4..........no

%e 9...5...10001...x^4 + 1......yes

%t fb[n_] := Block[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]],

%t t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k],

%t AppendTo[fr, 1]; t = t - Fibonacci[k],

%t AppendTo[fr, 0]]; k--]; fr]; t = Table[fb[n],

%t {n, 1, 350}];

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

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

%t Table[p[n, x], {n, 1, 40}] (* Zeckendorf polynomials *)

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

%t AppendTo[u, n]], {n, 300}]; u (* A207813 *)

%Y Cf. A206073, A206074.

%K nonn

%O 1,1

%A _Clark Kimberling_, Feb 20 2012

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Last modified June 17 00:17 EDT 2021. Contains 345080 sequences. (Running on oeis4.)