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

1,3

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.

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.

Example

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).

Prog

(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))

Crossrefs

Cf. A300332, A001348.

Sequence in context: A282516 A158092 A145264 * A357019 A352562 A362326

Adjacent sequences: A300330 A300331 A300332 * A300334 A300335 A300336

Keyword

nonn

Author

Peter Luschny, Mar 03 2018

Status

approved