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

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, 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, 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

Offset: 1

Author

Paolo P. Lava and Giorgio Balzarotti, Jun 05 2006

Keywords

• easy
• nonn

Comments

Any such k must be < n^2, if it exists. See comment in A379953. - Robert Israel, Jan 16 2025

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

Examples

			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.
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.
		

Links

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

Crossrefs

Cf. A071086, A379953 (largest solution for k, if it exists), A379954 (smallest solution, if it exists).

Programs

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

Extensions

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