cp's OEIS Frontend

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

%I A291850 #17 Jun 17 2025 05:17:07
%S A291850 8,256,1944,6655,8192,25000,35937,62208,134456,212960,262144,344605,
%T A291850 453962,472392,692759,800000,1149984,1288408,1617165,1990656,2970344,
%U A291850 4302592,6075000,6814720,8388608,8732691,11358856,14526784,15116544,19808792,20796875,22168288
%N A291850 Numbers k such that k^2 is the sum of two positive 5th powers.
%C A291850 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.
%C A291850 When k is in this sequence, k * (n^5) (n = 2, 3, ... ) is also in this sequence.
%e A291850 8^2 = 2^5 + 2^5, so 8 is in the sequence.
%e A291850 6655^2 = 22^5 + 33^5, so 6655 is in the sequence.
%t A291850 lst={};Do[If[IntegerQ[(n^2-a^5)^(1/5)],AppendTo[lst,n]],{n,9000},{a,(n^2/2)^(1/5)}]; lst
%o A291850 (PARI)
%o A291850 upto(n) = {
%o A291850     my(res = List(), u = n^2, i5);
%o A291850     for(i = 1, sqrtnint(u, 5),
%o A291850         i5 = i^5;
%o A291850         for(j = i, sqrtnint(u - i5, 5),
%o A291850             c = i5 + j^5;
%o A291850             if(issquare(c, &sc),
%o A291850                 listput(res, sc))));
%o A291850     Set(res)} \\ _David A. Corneth_, Jun 17 2025
%Y A291850 Cf. A000404, A004431, A003325, A050801, A003336, A003347.
%K A291850 nonn
%O A291850 1,1
%A A291850 _XU Pingya_, Sep 04 2017