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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.

%I A299964 #8 Apr 06 2024 14:57:32
%S A299964 19,39,97,147,247,259,327,399,410,427,481,650,777,890,903,1010,1027,
%T A299964 1130,1209,1267,1443,1490,1533,1677,1730,1767,1802,1813,1898,1911,
%U A299964 1970,2037,2119,2210,2330,2378,2667,2793,2847,3050,3170,3297,3367,3477,3530,3603
%N 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.
%C A299964 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.
%H A299964 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 A299964 (Julia)
%o A299964 function countA299928(n)
%o A299964     R, z = PolynomialRing(ZZ, "z")
%o A299964     K = Int(floor(5.383*log(n)^1.161)) # Bounds from
%o A299964     M = Int(floor(2*sqrt(n/3)))  # Fouvry & Levesque & Waldschmidt
%o A299964     N = QQ(n); count = 0
%o A299964     P(u) = (p for p in u:M if isprime(ZZ(p)))
%o A299964     for k in 3:K
%o A299964         e = Int(eulerphi(ZZ(k)))
%o A299964         c = cyclotomic(k, z)
%o A299964         for y in P(2), x in P(y+1)
%o A299964             if N == y^e*subst(c, QQ(x, y))
%o A299964                 count += 1
%o A299964     end end end
%o A299964     return count
%o A299964 end
%o A299964 A299964list(upto) = [n for n in 1:upto if countA299928(n) > 1]
%o A299964 println(A299964list(3640))
%Y A299964 Cf. A293654, A296095, A299214, A299498, A299733, A299928, A299929, A299930, A299956.
%K A299964 nonn
%O A299964 1,1
%A A299964 _Peter Luschny_, Feb 25 2018