A299964 Integers represented in more than one way by a cyclotomic binary form f(x,y) where x and y are prime numbers and 0 < y < x.

19, 39, 97, 147, 247, 259, 327, 399, 410, 427, 481, 650, 777, 890, 903, 1010, 1027, 1130, 1209, 1267, 1443, 1490, 1533, 1677, 1730, 1767, 1802, 1813, 1898, 1911, 1970, 2037, 2119, 2210, 2330, 2378, 2667, 2793, 2847, 3050, 3170, 3297, 3367, 3477, 3530, 3603

Offset: 1

Author

Peter Luschny, Feb 25 2018

Keywords

• nonn

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 in this sequence if f(x,y) = n has more than one integer solution where f is a cyclotomic binary form and x and y are prime numbers with 0 < y < x.

Links

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

Crossrefs

Cf. A293654, A296095, A299214, A299498, A299733, A299928, A299929, A299930, A299956.

Programs

Julia

function countA299928(n)
    R, z = PolynomialRing(ZZ, "z")
    K = Int(floor(5.383*log(n)^1.161)) # Bounds from
    M = Int(floor(2*sqrt(n/3)))  # Fouvry & Levesque & Waldschmidt
    N = QQ(n); count = 0
    P(u) = (p for p in u:M if isprime(ZZ(p)))
    for k in 3:K
        e = Int(eulerphi(ZZ(k)))
        c = cyclotomic(k, z)
        for y in P(2), x in P(y+1)
            if N == y^e*subst(c, QQ(x, y))
                count += 1
    end end end
    return count
end
A299964list(upto) = [n for n in 1:upto if countA299928(n) > 1]
println(A299964list(3640))