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

18, 36, 41, 46, 54, 58, 60, 72, 75, 76, 81, 82, 84, 87, 88, 90, 92, 96, 100, 108, 114, 116, 120, 123, 126, 132, 134, 138, 140, 142, 144, 145, 150, 152, 156, 159, 160, 162, 164, 168, 170, 171, 174, 176, 178, 180, 184, 185, 186, 189, 190, 192, 198, 200, 201, 202, 203

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..57.

Example

41 is in the sequence because 41^3 = 33^3 + 32^3 + 6^3 = 40^3 + 17^3 + 2^3.

Mathematica

q[k_] := Count[IntegerPartitions[k^3, {3}, Range[0, k-1]^3], _?(UnsameQ @@ # &)] > 1; Select[Range[200], q] (* Amiram Eldar, Sep 03 2021 *)

Prog

(Python)
from itertools import combinations
from collections import Counter
from sympy import integer_nthroot
def icuberoot(n): return integer_nthroot(n, 3)[0]
def aupto(kmax):
    cubes = [i**3 for i in range(kmax+1)]
    cands, cubesset = (sum(c) for c in combinations(cubes, 3)), set(cubes)
    c = Counter(s for s in cands if s in cubesset)
    return sorted(icuberoot(s) for s in c if c[s] >= 2)
print(aupto(203)) # Michael S. Branicky, Sep 04 2021

Crossrefs

Subsequence of A023042.

Cf. A001235, A346071.

Sequence in context: A256878 A285318 A040306 * A303282 A073332 A261991

Adjacent sequences: A346134 A346135 A346136 * A346138 A346139 A346140

Keyword

nonn

Author

Sebastian Magee, Jul 30 2021

Status

approved