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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
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
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] (* Jean-François 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
KEYWORD
nonn
AUTHOR
Michel Waldschmidt, Feb 16 2018
STATUS
approved