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A296095
Integers represented by cyclotomic binary forms.
15
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
KEYWORD
nonn
AUTHOR
Michel Waldschmidt, Feb 14 2018
STATUS
approved