

A299214


Number of representations of integers by cyclotomic binary forms.


8



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
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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: A073926 A073925 A053321 * A174256 A037239 A205869
Adjacent sequences: A299211 A299212 A299213 * A299215 A299216 A299217


KEYWORD

sign


AUTHOR

Michel Waldschmidt, Feb 16 2018


STATUS

approved



