

A299214


Number of representations of integers by cyclotomic binary forms.


10



0, 0, 8, 16, 8, 0, 24, 4, 16, 8, 8, 12, 40, 0, 0, 40, 16, 4, 24, 8, 24, 0, 0, 0, 24, 8, 12, 24, 8, 0, 32, 8, 0, 8, 0, 16, 32, 0, 24, 8, 8, 0, 32, 0, 8, 0, 0, 12, 40, 12, 0, 32, 8, 0, 8, 0, 32, 8, 0, 0, 48, 0, 24, 40, 16, 0, 24, 8, 0, 0, 0, 4, 48, 8, 12, 24
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


COMMENTS

a(m) is the number of solutions of the equation Phi_n(x,y) = m with n >= 3 and max{x,y} >= 2. Here the binary form Phi_n(x,y) is the homogeneous version of the cyclotomic polynomial phi_n(t).
One can prove that a(m) is always a multiple of 4.


LINKS

Michel Waldschmidt, Table of n, a(n) for n = 1..1000
Étienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.


MAPLE

x := 'x'; y := 'y':
with(numtheory): for n from 3 to 1000 do
F[n] := expand(y^phi(n)*cyclotomic(n, x/y)) od:
g := 0:
for m from 1 to 1000 do
for n from 3 to 60 do # For the bounds see the reference.
for x from 60 to 60 do
for y from 60 to 60 do
if F[n] = m and max(abs(x), abs(y)) > 1
then g := g+1 fi:
od:
od:
od: a[m] := g: print(m, a[m]): g := 0
od:


MATHEMATICA

For[n = 3, n <= 100, n++, F[n] = Expand[y^EulerPhi[n] Cyclotomic[n, x/y]]]; g = 0; For[m = 1, m <= 100, m++, For[n = 3, n <= 60, n++, For[x = 60, x <= 60, x++, For[y = 60, y <= 60, y++, If[F[n] == m && Max[Abs[x], Abs[y] ] > 1, g = g+1]]]]; a[m] = g; Print[m, " ", a[m]]; g = 0];
Array[a, 100] (* JeanFrançois Alcover, Dec 01 2018, from Maple *)


PROG

(Julia)
using Nemo
function countA296095(n)
if n < 3 return 0 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
N = QQ(n); count = 0
for k in 3:K
e = Int(eulerphi(ZZ(k)))
c = cyclotomic(k, x)
for m in 1:M, j in 0:M if max(j, m) > 1
N == m^e*subst(c, QQ(j, m)) && (count += 1)
end end end
4*count
end
A299214list(upto) = [countA296095(n) for n in 1:upto]
print(A299214list(76)) # Peter Luschny, Feb 25 2018


CROSSREFS

The sequence of indices m with a(m) != 0 is A296095.
The sequence of indices m with a(m) = 0 is A293654.
Sequence in context: A073925 A053321 A335772 * A174256 A037239 A205869
Adjacent sequences: A299211 A299212 A299213 * A299215 A299216 A299217


KEYWORD

nonn


AUTHOR

Michel Waldschmidt, Feb 16 2018


STATUS

approved



