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

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, 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, 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

Offset: 1

Author

Antti Karttunen, Jan 16 2025

Keywords

• nonn

Comments

For odd n, a(n) is even.

If n = p^j where p is a prime >= 5, then a(n) = 0 (see link for proof). - Robert Israel, Jan 16 2025

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Robert Israel, Proof of comment (which applies to A119612, A379953 and A379954)

Crossrefs

Cf. A119612, A379953.

Programs

Mathematica

A379954[n_] := Catch[Do[If[PowerMod[k*n, 3, # + k^3] == 0, Throw[k]], {k, n^2 - 1}] & [n^3]; 0];
Array[A379954, 100] (* Paolo Xausa, Jan 19 2025 *)

PARI

A379954(n) = { for(k=1,n^2, if(!(((n*k)^3)%(k^3+n^3)), return(k))); (0); };