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 A296095 Integers represented by cyclotomic binary forms. 12
 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 25, 26, 27, 28, 29, 31, 32, 34, 36, 37, 39, 40, 41, 43, 45, 48, 49, 50, 52, 53, 55, 57, 58, 61, 63, 64, 65, 67, 68, 72, 73, 74, 75, 76, 79, 80, 81, 82, 84, 85, 89, 90, 91, 93, 97, 98, 100, 101, 103, 104, 106, 108, 109, 111, 112, 113, 116, 117, 121, 122 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 LINKS Michel Waldschmidt and Peter Luschny (Michel Waldschmidt to 519), Table of n, a(n) for n = 1..1000 Etienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017. MAPLE with(numtheory): for n from 3 to 1000 do F[n] := expand(y^phi(n)*cyclotomic(n, x/y)) od: for m to 1000 do for n from 3 to 50 do for x from -50 to 50 do for y from -50 to 50 do if `and`(F[n] = m, max(abs(x), abs(y)) > 1) then print(m); m := m+1; n := 3; x := -50; y := -50 end if end do end do end do end do; MATHEMATICA isA296095[n_]:= If[n<3, Return[False], logn = Log[n]^1.161; K = Floor[5.383*logn]; M = Floor[2*(n/3)^(1/2)]; k = 3; While[True,    If[k==7,       K = Ceiling[4.864*logn];       M = Ceiling[2*(n/11)^(1/4)]    ];    For[y=2, y<=M, y++,       p[z_] = y^EulerPhi[k]*Cyclotomic[k, z];       For[x=1, x<=y, x++, If[n==p[x/y], Return[True]]]    ];    k++;    If[k>K, Break[]] ]; Return[False] ]; Select[Range[122], isA296095] (* Jean-François Alcover, Feb 20 2018, translated from Peter Luschny's Sage script, updated Mar 01 2018 *) PROG (Sage) var('z') def isA296095(n):     if n < 3: return false     logn = log(n)^1.161     K = floor(5.383*logn)     M = floor(2*(n/3)^(1/2))     k = 3     while true:         if k == 7:             K = ceil(4.864*logn)             M = ceil(2*(n/11)^(1/4))         for y in (2..M):             p = y^euler_phi(k)*cyclotomic_polynomial(k, z)             for x in (1..y):                 if n == p.substitute(z = x/y): return true         k += 1         if k > K: break     return false def A296095list(upto):     return [n for n in (1..upto) if isA296095(n)] print(A296095list(122)) # Peter Luschny, Feb 28 2018 (Julia) using Nemo function isA296095(n)     n < 3 && return false     R, z = PolynomialRing(ZZ, "z")     N = QQ(n)     # Bounds from Fouvry & Levesque & Waldschmidt     logn = log(n)^1.161     K = Int(floor(5.383*logn))     M = Int(floor(2*(n/3)^(1/2)))     k = 3     while true         c = cyclotomic(k, z)         e = Int(eulerphi(ZZ(k)))         if k == 7             K = Int(ceil(4.864*logn))             M = Int(ceil(2*(n/11)^(1/4)))         end         for y in 2:M, x in 1:y             N == y^e*subst(c, QQ(x, y)) && return true         end         k += 1         k > K && break     end     return false end A296095list(upto) = [n for n in 1:upto if isA296095(n)] println(A296095list(2040)) # Peter Luschny, Feb 28 2018 CROSSREFS Complement of A293654. Sequence in context: A063992 A171520 A039233 * A075748 A039177 A058986 Adjacent sequences:  A296092 A296093 A296094 * A296096 A296097 A296098 KEYWORD nonn AUTHOR Michel Waldschmidt, Feb 14 2018 STATUS approved

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Last modified December 9 20:24 EST 2018. Contains 318023 sequences. (Running on oeis4.)