A300333 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.

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, 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, 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

Offset: 1

Author

Peter Luschny, Mar 03 2018

Keywords

• nonn

Comments

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

The indices of the nonzero values are in A300332.

Examples

			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
a(49) = 3 because 49 = f(5, 3).
a(217) = 3 because 217 = f(13, 3).
a(448) = 7 because 448 = g(2, 2).
a(2047) = 11 because 2047 = h(2, 1).
		

Links

• Peter Luschny, Table of n, a(n) for n = 1..10000
• Étienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.

Crossrefs

Cf. A300332, A001348.

Programs

Julia

using Primes, Nemo
function A300333(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)))
    k, p = 2, 0
    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
            N == y^e*subst(c, QQ(x,y)) &&  (p = k)
        end
        k = nextprime(k+1)
    end
    return p
end
A300333list(upto) = [A300333(n) for n in 1:upto]
println(A300333list(121))