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A346137 Numbers k such that k^3 = x^3 + y^3 + z^3, x > y > z >= 0, has at least 2 distinct solutions.

%I A346137 #63 Oct 05 2021 22:46:40
%S A346137 18,36,41,46,54,58,60,72,75,76,81,82,84,87,88,90,92,96,100,108,114,
%T A346137 116,120,123,126,132,134,138,140,142,144,145,150,152,156,159,160,162,
%U A346137 164,168,170,171,174,176,178,180,184,185,186,189,190,192,198,200,201,202,203
%N A346137 Numbers k such that k^3 = x^3 + y^3 + z^3, x > y > z >= 0, has at least 2 distinct solutions.
%C A346137 This sequence is based on a generalization of Fermat's last theorem with n=3, in which three terms are added. Fermat's Theorem states that there are no solution with only two terms, this sequence shows there are many integers for which there are multiple solutions if three terms are allowed. The sequence is also related to the Taxicab numbers.
%e A346137 41 is in the sequence because 41^3 = 33^3 + 32^3 + 6^3 = 40^3 + 17^3 + 2^3.
%t A346137 q[k_] := Count[IntegerPartitions[k^3, {3}, Range[0, k-1]^3], _?(UnsameQ @@ # &)] > 1; Select[Range[200], q] (* _Amiram Eldar_, Sep 03 2021 *)
%o A346137 (Python)
%o A346137 from itertools import combinations
%o A346137 from collections import Counter
%o A346137 from sympy import integer_nthroot
%o A346137 def icuberoot(n): return integer_nthroot(n, 3)[0]
%o A346137 def aupto(kmax):
%o A346137     cubes = [i**3 for i in range(kmax+1)]
%o A346137     cands, cubesset = (sum(c) for c in combinations(cubes, 3)), set(cubes)
%o A346137     c = Counter(s for s in cands if s in cubesset)
%o A346137     return sorted(icuberoot(s) for s in c if c[s] >= 2)
%o A346137 print(aupto(203)) # _Michael S. Branicky_, Sep 04 2021
%Y A346137 Subsequence of A023042.
%Y A346137 Cf. A001235, A346071.
%K A346137 nonn
%O A346137 1,1
%A A346137 _Sebastian Magee_, Jul 30 2021