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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: A073925 A053321 A335772 * 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 September 20 17:16 EDT 2020. Contains 337265 sequences. (Running on oeis4.)