A300333 - OEIS

%I #11 Mar 07 2018 15:58:12

%S 0,0,2,2,0,0,3,0,0,0,0,3,3,0,0,0,0,0,3,0,3,0,0,0,0,0,3,3,0,0,5,0,0,0,

%T 0,0,3,0,3,0,0,0,3,0,0,0,0,3,3,0,0,3,0,0,0,0,3,0,0,0,3,0,3,0,0,0,3,0,

%U 0,0,0,0,3,0,3,3,0,0,3,5,0,0,0,3,0,0,0,0,0,0,3,0,3,0,0,0,3,0,0,0,0,0,3,0,0,0,0,3,3,0,3,3,0,0,0,0,3,0,0,0,5

%N a(n) = max{ p prime | n = Sum_{j in 0:p-1} x^j*y^(p-j-1)} where x and y are positive integers with max(x, y) >= 2 or 0 if no such representation exists.

%C All prime numbers appear as values. The earliest appearance of the prime p has the index 2^p - 1 (Mersenne number).

%C The indices of the nonzero values are in A300332.

%H Peter Luschny, <a href="/A300333/b300333.txt">Table of n, a(n) for n = 1..10000</a>

%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 Let f(x,y) = y^2 + x*y + x^2, g(x,y) = y^6 + x*y^5 + x^2*y^4 + x^3*y^3 + x^4*y^2 + x^5*y + x^6 and h(x,y) = Sum_{j in 0:10} x^j*y^(10-j). Then

%e a(49) = 3 because 49 = f(5, 3).

%e a(217) = 3 because 217 = f(13, 3).

%e a(448) = 7 because 448 = g(2, 2).

%e a(2047) = 11 because 2047 = h(2, 1).

%o (Julia)

%o using Primes, Nemo

%o function A300333(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     k, p = 2, 0

%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             N == y^e*subst(c, QQ(x,y)) &&  (p = k)

%o         end

%o         k = nextprime(k+1)

%o     end

%o     return p

%o end

%o A300333list(upto) = [A300333(n) for n in 1:upto]

%o println(A300333list(121))

%Y Cf. A300332, A001348.

%K nonn

%O 1,3

%A _Peter Luschny_, Mar 03 2018