A299498 - OEIS

%I #9 Apr 06 2024 15:00:17

%S 3,5,7,10,11,13,17,19,21,25,26,29,31,34,37,39,41,43,49,50,53,55,57,58,

%T 61,65,67,73,74,79,82,85,89,91,93,97,101,103,106,109,111,113,121,122,

%U 125,127,129,130,133,137,139,145,146,147,149,151,157,163,169,170

%N Integers primitively represented by cyclotomic binary forms.

%C 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.

%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.

%o (Julia)

%o using Nemo

%o function isA299498(n)

%o     isPrimeTo(n, k) = gcd(ZZ(n), ZZ(k)) == ZZ(1)

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

%o     K = Int(floor(5.383*log(n)^1.161)) # Bounds from

%o     M = Int(floor(2*sqrt(n/3)))  # Fouvry & Levesque & Waldschmidt

%o     N = QQ(n)

%o     for k in 3:K

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

%o         c = cyclotomic(k, x)

%o         for m in 1:M, j in m+1:M if isPrimeTo(m, j)

%o             N == m^e*subst(c, QQ(j,m)) && return true

%o     end end end

%o     return false

%o end

%o A299498list(upto) = [n for n in 1:upto if isA299498(n)]

%o print(A299498list(170))

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

%K nonn

%O 1,1

%A _Peter Luschny_, Feb 25 2018