A291850 Numbers k such that k^2 is the sum of two positive 5th powers.

8, 256, 1944, 6655, 8192, 25000, 35937, 62208, 134456, 212960, 262144, 344605, 453962, 472392, 692759, 800000, 1149984, 1288408, 1617165, 1990656, 2970344, 4302592, 6075000, 6814720, 8388608, 8732691, 11358856, 14526784, 15116544, 19808792, 20796875, 22168288

Offset

1,1

Comments

If a^5 + b^5 = m, then (ma)^5 + (mb)^5 = m^6 = (m^3)^2 is square. Therefore A003347(n)^3 are terms of this sequence.

When k is in this sequence, k * (n^5) (n = 2, 3, ... ) is also in this sequence.

Links

Table of n, a(n) for n=1..32.

Example

8^2 = 2^5 + 2^5, so 8 is in the sequence.
6655^2 = 22^5 + 33^5, so 6655 is in the sequence.

Mathematica

lst={}; Do[If[IntegerQ[(n^2-a^5)^(1/5)], AppendTo[lst, n]], {n, 9000}, {a, (n^2/2)^(1/5)}]; lst

Prog

(PARI)
upto(n) = {
    my(res = List(), u = n^2, i5);
    for(i = 1, sqrtnint(u, 5),
        i5 = i^5;
        for(j = i, sqrtnint(u - i5, 5),
            c = i5 + j^5;
            if(issquare(c, &sc),
                listput(res, sc))));
    Set(res)} \\ David A. Corneth, Jun 17 2025

Crossrefs

Cf. A000404, A004431, A003325, A050801, A003336, A003347.

Sequence in context: A300176 A372670 A387611 * A130034 A128787 A013824

Adjacent sequences: A291847 A291848 A291849 * A291851 A291852 A291853

Keyword

nonn

Author

XU Pingya, Sep 04 2017

Status

approved