cp's OEIS Frontend

A119612 Number of integers of the form (nk)^3/(n^3+k^3), k>=1.

%I A119612 #23 Jan 18 2025 19:31:58
%S A119612 0,1,1,1,0,3,0,1,1,1,0,3,0,2,1,1,0,3,0,1,1,1,0,3,0,1,1,2,0,3,0,1,1,1,
%T A119612 0,3,0,1,1,1,0,6,0,1,1,1,0,3,0,1,1,1,0,3,0,2,2,1,0,3,0,1,1,1,1,3,0,1,
%U A119612 1,4,0,3,0,1,1,1,0,5,0,1,1,1,0,7,0,2,1,1,0,3,0,1,1,1,0,3,0,2,1,1,0,3,0,1,3
%N A119612 Number of integers of the form (nk)^3/(n^3+k^3), k>=1.
%C A119612 Any such k must be < n^2, if it exists. See comment in A379953. - _Robert Israel_, Jan 16 2025
%C A119612 If n = p^j where p is a prime >= 5, then a(n) = 0 (see link for proof). - _Robert Israel_, Jan 16 2025
%H A119612 Antti Karttunen, <a href="/A119612/b119612.txt">Table of n, a(n) for n = 1..10000</a>
%H A119612 Robert Israel, <a href="/A379953/a379953.pdf">Proof of the latter comment</a> (which applies to A119612, A379953 and A379954)
%e A119612 a(6) = 3 as there are the following three solutions: (1) n=6, k=3: (6^3 * 3^3) / (6^3 + 3^3) = 5832/243 = 24, (2) n=6, k=6: ? (6^3 * 6^3) / (6^3 + 6^3) = 46656/432 = 108, and (3) n=6, k=12: (6^3 * 12^3) / (6^3 + 12^3) = 373248/1944 = 192.
%e A119612 a(14) = 2 as there are two solutions, (1) n=14, k=14: (14^3 * 14^3) / (14^3 + 14^3) = 14^6 / (2 * 14^3) = 1372 and (2) n=14, k=42: (14^3 * 42^3)/(14^3 + 42^3)= (2744 * 74088)/(2744 + 74088) = 203297472 / 76832 = 2646.
%o A119612 (PARI) A119612(n) = sum(k=1, n^2, !(((n*k)^3)%(k^3+n^3))); \\ (after PARI-code in A071086) - _Antti Karttunen_, Jan 16 2025
%Y A119612 Cf. A071086, A379953 (largest solution for k, if it exists), A379954 (smallest solution, if it exists).
%K A119612 easy,nonn
%O A119612 1,6
%A A119612 _Paolo P. Lava_ and _Giorgio Balzarotti_, Jun 05 2006
%E A119612 Corrected offset (from 0 to 1), terms a(70) (from 3 to 4) and a(78) (from 4 to 5), and extended the data section to 105 terms - _Antti Karttunen_, Jan 16 2025