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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 (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

sign

AUTHOR

Michel Waldschmidt, Feb 16 2018

STATUS

approved

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Last modified January 20 13:11 EST 2019. Contains 319332 sequences. (Running on oeis4.)