|
|
A206942
|
|
Numbers of the form Phi_k(m) with k > 2 and |m| > 1.
|
|
5
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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. A006511 for phiinv function in the Mathematica program.
|
|
KEYWORD
|
nonn,changed
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|