A299498 Integers primitively represented by cyclotomic binary forms.

3, 5, 7, 10, 11, 13, 17, 19, 21, 25, 26, 29, 31, 34, 37, 39, 41, 43, 49, 50, 53, 55, 57, 58, 61, 65, 67, 73, 74, 79, 82, 85, 89, 91, 93, 97, 101, 103, 106, 109, 111, 113, 121, 122, 125, 127, 129, 130, 133, 137, 139, 145, 146, 147, 149, 151, 157, 163, 169, 170

Offset

1,1

Comments

A cyclotomic binary form over Z is a homogeneous polynomial in two variables which has the form f(x, y) = y^EulerPhi(k)*CyclotomicPolynomial(k, x/y) where k is some integer >= 3. An integer n is primitively represented by f if f(x,y) = n has an integer solution such that x is prime to y.

Links

Table of n, a(n) for n=1..60.

Étienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.

Prog

(Julia)
using Nemo
function isA299498(n)
    isPrimeTo(n, k) = gcd(ZZ(n), ZZ(k)) == ZZ(1)
    R, x = PolynomialRing(ZZ, "x")
    K = Int(floor(5.383*log(n)^1.161)) # Bounds from
    M = Int(floor(2*sqrt(n/3)))  # Fouvry & Levesque & Waldschmidt
    N = QQ(n)
    for k in 3:K
        e = Int(eulerphi(ZZ(k)))
        c = cyclotomic(k, x)
        for m in 1:M, j in m+1:M if isPrimeTo(m, j)
            N == m^e*subst(c, QQ(j, m)) && return true
    end end end
    return false
end
A299498list(upto) = [n for n in 1:upto if isA299498(n)]
print(A299498list(170))

Crossrefs

Cf. A293654, A296095, A299214, A299733, A299928, A299929, A299930, A299956, A299964.

Sequence in context: A190345 A189516 A138968 * A206942 A189171 A189220

Adjacent sequences: A299495 A299496 A299497 * A299499 A299500 A299501

Keyword

nonn

Author

Peter Luschny, Feb 25 2018

Status

approved