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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 (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] (* 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: A073926 A073925 A053321 * A174256 A037239 A205869

Adjacent sequences:  A299211 A299212 A299213 * A299215 A299216 A299217

KEYWORD

nonn

AUTHOR

Michel Waldschmidt, Feb 16 2018

STATUS

approved

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Last modified October 14 02:29 EDT 2019. Contains 327995 sequences. (Running on oeis4.)