

A300331


Integers represented by a cyclotomic binary form Phi{k}(x,y) with positive integers x and y where max(x, y) >= 2 and the index k is not prime.


0



5, 8, 9, 10, 11, 16, 17, 18, 20, 25, 26, 29, 32, 34, 36, 40, 41, 45, 50, 53, 55, 58, 64, 65, 68, 72, 74, 81, 82, 85, 89, 90, 98, 100, 101, 104, 106, 113, 116, 122, 125, 128, 130, 136, 137, 144, 145, 146, 149, 153, 160, 162, 164, 170, 173, 176, 178, 180, 185
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OFFSET

1,1


COMMENTS

A cyclotomic binary form is a homogeneous polynomial in two variables of the form p(x, y) = y^phi(k)*Phi(k, x/y) where Phi(k, z) is a cyclotomic polynomial of index k and phi is Euler's totient function. An integer m is represented by p if p(x,y) = m has an integer solution.
m is in this sequence if and only if m is in A296095 but not in A300332. This means m can be represented by a cyclotomic binary form but not as m = Sum_{j in 0:p1} x^j*y^(pj1) with p prime.


LINKS

Table of n, a(n) for n=1..59.
Étienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.


EXAMPLE

1037 is in this sequence because 1037 = f(26,19) = f(29,14) with f(x,y) = y^2 + x^2 are the only representations of 1037 by a cyclotomic binary form (which has index 4).
1031 is not in this sequence because 1031 = f(5,2) where f(x,y) = x^4 + y*x^3 + y^2*x^2 + y^3*x + y^4 (which has index 5).


PROG

(Julia)
using Nemo
function isA300331(n)
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)))
r = false
k = 2
while k <= K
if k == 7
K = Int(ceil(4.864*logn))
M = Int(ceil(2*(n/11)^(1/4)))
end
e = Int(eulerphi(ZZ(k)))
c = cyclotomic(k, z)
for y in 2:M, x in 1:y
if N == y^e*subst(c, QQ(x, y))
isprime(ZZ(k)) && return false
r = true
end
end
k += 1
end
return r
end
A300331list(upto) = [n for n in 1:upto if isA300331(n)]
println(A300331list(185))


CROSSREFS

Cf. A296095, A293654, A299930, A300332.
Sequence in context: A205862 A293795 A165991 * A293271 A323215 A334818
Adjacent sequences: A300328 A300329 A300330 * A300332 A300333 A300334


KEYWORD

nonn


AUTHOR

Peter Luschny, Mar 06 2018


STATUS

approved



