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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. 8
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 (list; graph; refs; listen; history; text; internal format)
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 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

Table of n, a(n) for n=1..46.

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

PROG

(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))

CROSSREFS

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

Sequence in context: A258016 A217731 A195048 * A041714 A041716 A041718

Adjacent sequences:  A299961 A299962 A299963 * A299965 A299966 A299967

KEYWORD

nonn

AUTHOR

Peter Luschny, Feb 25 2018

STATUS

approved

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Last modified January 24 07:18 EST 2020. Contains 331189 sequences. (Running on oeis4.)