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A379954 Smallest positive k such that (n*k)^3/(n^3+k^3) is an integer, or 0 if no such k exists.

%I A379954 #16 Jan 19 2025 16:16:52
%S A379954 0,2,6,4,0,3,0,8,18,10,0,6,0,14,30,16,0,9,0,20,42,22,0,12,0,26,54,28,
%T A379954 0,15,0,32,66,34,0,18,0,38,78,40,0,14,0,44,90,46,0,24,0,50,102,52,0,
%U A379954 27,0,56,114,58,0,30,0,62,126,64,260,33,0,68,138,70,0,36,0,74,150,76,0,39,0,80,162,82,0,28,0,86
%N A379954 Smallest positive k such that (n*k)^3/(n^3+k^3) is an integer, or 0 if no such k exists.
%C A379954 For odd n, a(n) is even.
%C A379954 If n = p^j where p is a prime >= 5, then a(n) = 0 (see link for proof). - _Robert Israel_, Jan 16 2025
%H A379954 Antti Karttunen, <a href="/A379954/b379954.txt">Table of n, a(n) for n = 1..10000</a>
%H A379954 Robert Israel, <a href="/A379953/a379953.pdf">Proof of comment</a> (which applies to A119612, A379953 and A379954)
%t A379954 A379954[n_] := Catch[Do[If[PowerMod[k*n, 3, # + k^3] == 0, Throw[k]], {k, n^2 - 1}] & [n^3]; 0];
%t A379954 Array[A379954, 100] (* _Paolo Xausa_, Jan 19 2025 *)
%o A379954 (PARI) A379954(n) = { for(k=1,n^2, if(!(((n*k)^3)%(k^3+n^3)), return(k))); (0); };
%Y A379954 Cf. A119612, A379953.
%K A379954 nonn
%O A379954 1,2
%A A379954 _Antti Karttunen_, Jan 16 2025