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A206942 Numbers of the form Phi_k(m) with k > 2 and |m| > 1. 5

%I #40 Feb 21 2018 15:58:44

%S 3,5,7,10,11,13,17,21,26,31,37,43,50,57,61,65,73,82,91,101,111,121,

%T 122,127,133,145,151,157,170,183,197,205,211,226,241,257,273,290,307,

%U 325,331,341,343,362,381,401,421,442,463,485,507,521,530,547,553

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

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

%C 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.

%C 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.

%H Etienne Fouvry, Claude Levesque, Michel Waldschmidt, <a href="https://arxiv.org/abs/1712.09019">Representation of integers by cyclotomic binary forms</a>, arXiv:1712.09019 [math.NT], 2017.

%e a(1) = 3 = Phi_6(2) = Cyclotomic(6,2).

%e a(2) = 5 = Phi_4(2) = Cyclotomic(4,2).

%e ...

%e a(15) = 61 = Phi_5(-3) = Cyclotomic(5,-3).

%t 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]

%t (* Alternatively: *)

%t isA206942[n_] := If[n < 3, Return[False],

%t K = Floor[5.383 Log[n]^1.161]; M = Floor[2 Sqrt[n/3]];

%t For[k = 3, k <= K, k++, For[x = 2, x <= M, x++,

%t If[n == Cyclotomic[k, x], Return[True]]]];

%t Return[False]

%t ]; Select[Range[555], isA206942] (* _Peter Luschny_, Feb 21 2018 *)

%o (Julia)

%o using Nemo

%o function isA206942(n)

%o if n < 3 return false end

%o R, x = PolynomialRing(ZZ, "x")

%o K = Int(floor(5.383*log(n)^1.161)) # Bounds from

%o M = Int(floor(2*sqrt(n/3))) # Fouvry & Levesque & Waldschmidt

%o for k in 3:K

%o c = cyclotomic(k, x)

%o for m in 2:M

%o n == subst(c, m) && return true

%o end

%o end

%o return false

%o end

%o L = [n for n in 1:553 if isA206942(n)]; print(L) # _Peter Luschny_, Feb 21 2018

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

%Y Cf. A006511 for phiinv function in the Mathematica program.

%K nonn

%O 1,1

%A _Lei Zhou_, Feb 13 2012

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Last modified March 28 14:13 EDT 2024. Contains 371254 sequences. (Running on oeis4.)