A296095 Integers represented by cyclotomic binary forms.

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

Offset

1,1

Links

Michel Waldschmidt and Peter Luschny (Michel Waldschmidt to 519), 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

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)
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)
            for x in (1..y):
                if n == p(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: A324540 A171520 A039233 * A075748 A039177 A058986

Adjacent sequences: A296092 A296093 A296094 * A296096 A296097 A296098

Keyword

nonn

Author

Michel Waldschmidt, Feb 14 2018

Status

approved