A300331 - OEIS

%I #16 Mar 09 2018 04:44:13

%S 5,8,9,10,11,16,17,18,20,25,26,29,32,34,36,40,41,45,50,53,55,58,64,65,

%T 68,72,74,81,82,85,89,90,98,100,101,104,106,113,116,122,125,128,130,

%U 136,137,144,145,146,149,153,160,162,164,170,173,176,178,180,185

%N 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.

%C 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.

%C 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.

%H Étienne Fouvry, Claude Levesque, Michel Waldschmidt, <a href="https://arxiv.org/abs/1712.09019">Representation of integers by cyclotomic binary forms</a>, arXiv:1712.09019 [math.NT], 2017.

%e 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).

%e 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).

%o (Julia)

%o using Nemo

%o function isA300331(n)

%o     R, z = PolynomialRing(ZZ, "z")

%o     N = QQ(n)

%o     # Bounds from Fouvry & Levesque & Waldschmidt

%o     logn = log(n)^1.161

%o     K = Int(floor(5.383*logn))

%o     M = Int(floor(2*(n/3)^(1/2)))

%o     r = false

%o     k = 2

%o     while k <= K

%o         if k == 7

%o             K = Int(ceil(4.864*logn))

%o             M = Int(ceil(2*(n/11)^(1/4)))

%o         end

%o             e = Int(eulerphi(ZZ(k)))

%o             c = cyclotomic(k, z)

%o             for y in 2:M, x in 1:y

%o                 if N == y^e*subst(c, QQ(x, y))

%o                     isprime(ZZ(k)) && return false

%o                     r = true

%o                 end

%o             end

%o         k += 1

%o     end

%o     return r

%o end

%o A300331list(upto) = [n for n in 1:upto if isA300331(n)]

%o println(A300331list(185))

%Y Cf. A296095, A293654, A299930, A300332.

%K nonn

%O 1,1

%A _Peter Luschny_, Mar 06 2018