A206942 Numbers of the form Phi_k(m) with k > 2 and |m| > 1.

3, 5, 7, 10, 11, 13, 17, 21, 26, 31, 37, 43, 50, 57, 61, 65, 73, 82, 91, 101, 111, 121, 122, 127, 133, 145, 151, 157, 170, 183, 197, 205, 211, 226, 241, 257, 273, 290, 307, 325, 331, 341, 343, 362, 381, 401, 421, 442, 463, 485, 507, 521, 530, 547, 553

Offset

1,1

Comments

Phi_k(m) denotes the k-th cyclotomic polynomial evaluated at m.

We can see that for any integer b, b = Phi_2(b-1). However, if we make k>2 and |m|>1, Phi(k,m) are always positive integers that do not traverse the positive integer set.

The Mathematica program can generate this sequence to arbitrary upper bound maxdata without user's chosen of parameters. The parameter determination part of this program is explained in A206864.

Links

Table of n, a(n) for n=1..55.

Étienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.

Example

a(1) = 3 = Phi_6(2) = Cyclotomic(6,2).
a(2) = 5 = Phi_4(2) = Cyclotomic(4,2).
...
a(15) = 61 = Phi_5(-3) = Cyclotomic(5,-3).

Mathematica

phiinv[n_, pl_] :=  Module[{i, p, e, pe, val}, If[pl == {}, Return[If[n == 1, {1}, {}]]]; val = {}; p = Last[pl]; For[e = 0; pe = 1, e == 0 || Mod[n, (p - 1) pe/p] == 0, e++; pe *= p, val = Join[val, pe*phiinv[If[e == 0, n, n*p/pe/(p - 1)], Drop[pl, -1]]]]; Sort[val]]; phiinv[n_] := phiinv[n, Select[1 + Divisors[n], PrimeQ]]; maxdata = 560; max =  Ceiling[(1 + Sqrt[1 + 4*(maxdata - 1)])/4]*2; eb =  2*Floor[(Log[2, maxdata])/2 + 0.5]; While[eg = phiinv[eb]; lu = Length[eg]; lu == 0, eb = eb + 2]; t = Select[Range[eg[[Length[eg]]]], EulerPhi[#] <= eb &]; ap = SortBy[t, Cyclotomic[#, 2] &]; an =  SortBy[t, Cyclotomic[#, -2] &]; a = {}; Do[i = 2; While[i++; cc = Cyclotomic[ap[[i]], m]; cc <= maxdata, a = Append[a, cc]]; i = 2;  While[i++; cc = Cyclotomic[an[[i]], -m]; cc <= maxdata, a = Append[a, cc]], {m, 2, max}]; Union[a]
(* Alternatively: *)
isA206942[n_] := If[n < 3, Return[False],
    K = Floor[5.383 Log[n]^1.161]; M = Floor[2 Sqrt[n/3]];
    For[k = 3, k <= K, k++, For[x = 2, x <= M, x++,
        If[n == Cyclotomic[k, x], Return[True]]]];
    Return[False]
]; Select[Range[555], isA206942] (* Peter Luschny, Feb 21 2018 *)

Prog

(Julia)
using Nemo
function isA206942(n)
    if n < 3 return false end
    R, x = PolynomialRing(ZZ, "x")
    K = Int(floor(5.383*log(n)^1.161)) # Bounds from
    M = Int(floor(2*sqrt(n/3)))        # Fouvry & Levesque & Waldschmidt
    for k in 3:K
        c = cyclotomic(k, x)
        for m in 2:M
            n == subst(c, m) && return true
        end
    end
    return false
end
L = [n for n in 1:553 if isA206942(n)]; print(L) # Peter Luschny, Feb 21 2018

Crossrefs

Cf. A206225, A206710, A194712, A206292, A206864.

Cf. A006511 for phiinv function in the Mathematica program.

Sequence in context: A189516 A138968 A299498 * A189171 A189220 A189009

Adjacent sequences: A206939 A206940 A206941 * A206943 A206944 A206945

Keyword

nonn

Author

Lei Zhou, Feb 13 2012

Status

approved