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
KEYWORD
nonn
AUTHOR
Michel Waldschmidt, Feb 14 2018
STATUS
approved