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A377444 a(n) is the smallest integer k such that the Diophantine equation x^3 + y^3 + z^3 = k^3, where 0 < x <= y <= z has exactly n integer solutions.

%I A377444 #26 Jun 12 2025 13:00:05
%S A377444 6,18,54,87,108,216,174,348,396,324,696,864,492,1080,984,1728,1584,
%T A377444 1296,2160,1440,3312,3132,2880,2592,4176,4230,6624,3960,5184,6264,
%U A377444 4320,5760,6480,7200,10200,7920,9936,5940,8640,12060,11520,9900,14256,14400,16560,14760,15660,22140
%N A377444 a(n) is the smallest integer k such that the Diophantine equation x^3 + y^3 + z^3 = k^3, where 0 < x <= y <= z has exactly n integer solutions.
%C A377444 All the terms seem to be multiple of 3.
%H A377444 Chai Wah Wu, <a href="/A377444/b377444.txt">Table of n, a(n) for n = 1..120</a> (terms 1..66 from Zhining Yang)
%e A377444 a(2)=18, because 18^3 = 9^3 + 12^3 + 15^3 = 2^3 + 12^3 + 16^3 and no integer less than 18 has 2 solutions.
%t A377444 a = Table[SelectFirst[Table[{k, Length@Select[PowersRepresentations[k^3, 3, 3], #[[1]] > 0 &]}, {k, 3, 500, 3}], #[[2]] == k &], {k, 10}]
%o A377444 (Python)
%o A377444 from itertools import count
%o A377444 from sympy.solvers.diophantine.diophantine import power_representation
%o A377444 def A377444(n): return next(filter(lambda k:len(list(power_representation(k**3,3,3)))==n,count(1))) # _Chai Wah Wu_, Nov 19 2024
%Y A377444 Cf. A316359.
%K A377444 nonn,easy
%O A377444 1,1
%A A377444 _Zhining Yang_, Oct 28 2024