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

Author

XU Pingya, Sep 04 2017

Keywords

• nonn

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.

Examples

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

Crossrefs

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

Programs

Mathematica

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

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