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

%I A206942 #44 Apr 06 2024 15:04:59
%S A206942 3,5,7,10,11,13,17,21,26,31,37,43,50,57,61,65,73,82,91,101,111,121,
%T A206942 122,127,133,145,151,157,170,183,197,205,211,226,241,257,273,290,307,
%U A206942 325,331,341,343,362,381,401,421,442,463,485,507,521,530,547,553
%N A206942 Numbers of the form Phi_k(m) with k > 2 and |m| > 1.
%C A206942 Phi_k(m) denotes the k-th cyclotomic polynomial evaluated at m.
%C A206942 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 A206942 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 A206942 Étienne 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 A206942 a(1) = 3 = Phi_6(2) = Cyclotomic(6,2).
%e A206942 a(2) = 5 = Phi_4(2) = Cyclotomic(4,2).
%e A206942 ...
%e A206942 a(15) = 61 = Phi_5(-3) = Cyclotomic(5,-3).
%t A206942 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 A206942 (* Alternatively: *)
%t A206942 isA206942[n_] := If[n < 3, Return[False],
%t A206942     K = Floor[5.383 Log[n]^1.161]; M = Floor[2 Sqrt[n/3]];
%t A206942     For[k = 3, k <= K, k++, For[x = 2, x <= M, x++,
%t A206942         If[n == Cyclotomic[k, x], Return[True]]]];
%t A206942     Return[False]
%t A206942 ]; Select[Range[555], isA206942] (* _Peter Luschny_, Feb 21 2018 *)
%o A206942 (Julia)
%o A206942 using Nemo
%o A206942 function isA206942(n)
%o A206942     if n < 3 return false end
%o A206942     R, x = PolynomialRing(ZZ, "x")
%o A206942     K = Int(floor(5.383*log(n)^1.161)) # Bounds from
%o A206942     M = Int(floor(2*sqrt(n/3)))        # Fouvry & Levesque & Waldschmidt
%o A206942     for k in 3:K
%o A206942         c = cyclotomic(k, x)
%o A206942         for m in 2:M
%o A206942             n == subst(c, m) && return true
%o A206942         end
%o A206942     end
%o A206942     return false
%o A206942 end
%o A206942 L = [n for n in 1:553 if isA206942(n)]; print(L) # _Peter Luschny_, Feb 21 2018
%Y A206942 Cf. A206225, A206710, A194712, A206292, A206864.
%Y A206942 Cf. A006511 for phiinv function in the Mathematica program.
%K A206942 nonn
%O A206942 1,1
%A A206942 _Lei Zhou_, Feb 13 2012