A300332 -id:A300332 - cp's OEIS frontend

A299795 Numbers of the form p*2^(p-1) where p is prime.

4, 12, 80, 448, 11264, 53248, 1114112, 4980736, 96468992, 7784628224, 33285996544, 2542620639232, 45079976738816, 189115999977472, 3307330976350208, 238690780250636288, 17005592192950992896, 70328211781017665536, 4943727411754159833088, 83822005070936202543104

Offset: 1

Peter Luschny, Mar 03 2018

nonn

G. C. Greubel, Table of n, a(n) for n = 1..460

A subsequence of A001787 and A300332.

[NthPrime(n)*2^(NthPrime(n) -1): n in [1..30]]; // G. C. Greubel, Mar 07 2018

Primes := select(isprime, [$1..71]):
seq(p*2^(p-1), p in Primes);

Table[Prime[n]*2^(Prime[n] -1), {n,1,30}] (* G. C. Greubel, Mar 07 2018 *)

a(n) = my(p = prime(n)); p*2^(p-1); \\ Michel Marcus, Mar 07 2018

From Michel Marcus, Mar 07 2018: (Start)
a(n) = prime(n)*2^(prime(n)-1).
a(n) = A000040(n)*A061286(n).
a(n) = A001787(A000040(n)).
(End)

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.

Original entry on oeis.org

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

Peter Luschny, Mar 03 2018

nonn

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.

			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).

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.

Cf. A300332, A001348.

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))

A300331 Integers represented by a cyclotomic binary form Phi{k}(x,y) with positive integers x and y where max(x, y) >= 2 and the index k is not prime.

Original entry on oeis.org

5, 8, 9, 10, 11, 16, 17, 18, 20, 25, 26, 29, 32, 34, 36, 40, 41, 45, 50, 53, 55, 58, 64, 65, 68, 72, 74, 81, 82, 85, 89, 90, 98, 100, 101, 104, 106, 113, 116, 122, 125, 128, 130, 136, 137, 144, 145, 146, 149, 153, 160, 162, 164, 170, 173, 176, 178, 180, 185

Offset: 1

Peter Luschny, Mar 06 2018

nonn

A cyclotomic binary form is a homogeneous polynomial in two variables of the form p(x, y) = y^phi(k)*Phi(k, x/y) where Phi(k, z) is a cyclotomic polynomial of index k and phi is Euler's totient function. An integer m is represented by p if p(x,y) = m has an integer solution.

m is in this sequence if and only if m is in A296095 but not in A300332. This means m can be represented by a cyclotomic binary form but not as m = Sum_{j in 0:p-1} x^j*y^(p-j-1) with p prime.

			1037 is in this sequence because 1037 = f(26,19) = f(29,14) with f(x,y) = y^2 + x^2 are the only representations of 1037 by a cyclotomic binary form (which has index 4).
1031 is not in this sequence because 1031 = f(5,2) where f(x,y) = x^4 + y*x^3 + y^2*x^2 + y^3*x + y^4 (which has index 5).

Étienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.

Cf. A296095, A293654, A299930, A300332.

using Nemo
function isA300331(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)))
    r = false
    k = 2
    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
                if N == y^e*subst(c, QQ(x, y))
                    isprime(ZZ(k)) && return false
                    r = true
                end
            end
        k += 1
    end
    return r
end
A300331list(upto) = [n for n in 1:upto if isA300331(n)]
println(A300331list(185))

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A299795 Numbers of the form p*2^(p-1) where p is prime.

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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.

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A300331 Integers represented by a cyclotomic binary form Phi{k}(x,y) with positive integers x and y where max(x, y) >= 2 and the index k is not prime.

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