A206443 Least n such that L(n)<-1 and L(n)>L(n-1), where L(k) means the least root of the polynomial p(k,x) defined at A206284, and a(1)=13.

13, 37, 145, 157, 181, 517, 565, 661, 2101, 2197, 2581, 2773, 8725, 8917, 10357, 10453, 10837, 35029, 35413, 41173, 41557, 43093, 43861

Offset

1,1

Comments

A206074 gives an ordering {p(n,x)} of the polynomials with coefficients in {0,1}.

The least n for which p(n,x) has a root r less than -1 is 13, hence the choice of 13 as the initial term of A206443. (Specifically, p(13,x)=1+x^2+x^3, and r=-1.46557...) The next p(n,x) having a root less than -1 and >r is p(37,x)=1+x^2+x^5, with least root -1.1938...

Links

Table of n, a(n) for n=1..23.

Mathematica

highs := {First /@ #, Most[FoldList[Plus, 1, Length /@ #]]} &[Split[Rest[FoldList[Max, -\[Infinity], #]]]] &
f[polyInX_] := {Min[#], Max[#]} &[
  Map[#[[1]] &, DeleteCases[Map[{#, Head[#]} &, Chop[N[x /. Solve[polyInX == 0, x], 40]]], {_, Complex}]]]
t = Table[IntegerDigits[n, 2], {n, 1, 100000}];
b[n_] := Reverse[Array[x^(# - 1) &, {n + 1}]]
p[n_] := t[[n]].b[-1 + Length[t[[n]]]]
Table[p[n], {n, 1, 25}]
fitCriterion = Intersection[Map[#[[1]] &, DeleteCases[
       Table[{n, Boole[IrreduciblePolynomialQ[p[n]]]}, {n, 1, #}], {_, 0}]], Map[#[[1]] &, DeleteCases[
       Table[{n, CountRoots[#, {x, -Infinity, 0}] -
       CountRoots[#, {x, -1, 0}] &[p[n]]}, {n, 1, #}],
           {_, 0}]]] &[Length[t]];
polyNum = Map[{f[p[#]][[1]], #} &, fitCriterion];
up = Map[polyNum[[#]] &, highs[Map[#[[1]] &, polyNum]][[2]]]
down = Map[polyNum[[#]] &, highs[Map[#[[1]] &, -polyNum]][[2]]]
Table[up[[k, 2]], {k, 1, Length[up]}]      (* A206443 *)
Table[down[[k, 2]], {k, 1, Length[down]}]  (* A206444 *)
(* Peter J. C. Moses, Feb 06 2012 *)

Crossrefs

Cf. A206074, A206284, A206444.

Sequence in context: A155300 A155252 A265653 * A155297 A181151 A155903

Adjacent sequences: A206440 A206441 A206442 * A206444 A206445 A206446

Keyword

nonn,more

Author

Clark Kimberling, Feb 07 2012

Status

approved