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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 (list; graph; refs; listen; history; text; internal format)
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:p-1} x^j*y^(p-j-1) with p prime.

LINKS

Table of n, a(n) for n=1..59.

√Čtienne 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

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Last modified September 19 17:56 EDT 2020. Contains 337180 sequences. (Running on oeis4.)