A071086 -id:A071086 - cp's OEIS frontend

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

Paolo P. Lava and Giorgio Balzarotti, Jun 05 2006

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

			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.

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)

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

A119612(n) = sum(k=1, n^2, !(((n*k)^3)%(k^3+n^3))); \\ (after PARI-code in A071086) - Antti Karttunen, Jan 16 2025

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

A152492 a(n) = number of integers of the form (n*k)^2/(k^2 - n^2).

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 0, 1, 1, 1, 0, 8, 0, 0, 4, 1, 0, 2, 0, 4, 3, 0, 0, 9, 0, 0, 1, 2, 0, 7, 0, 1, 2, 0, 1, 8, 0, 0, 1, 4, 0, 5, 0, 1, 5, 0, 0, 9, 0, 1, 1, 1, 0, 2, 1, 4, 1, 0, 0, 23, 0, 0, 3, 1, 1, 4, 0, 1, 1, 2, 0, 10, 0, 0, 4, 1, 0, 4, 0, 4, 1, 0, 0, 17, 0, 0, 1, 1, 0, 8

Offset: 1

Ctibor O. Zizka, Dec 06 2008

k needs to be checked only up through n^2+1 since beyond this n^2 < (n*k)^2/(k^2 - n^2) < n^2 + 1 and thus can't be an integer. - Micah Manary, Aug 27 2022

Micah Manary, Table of n, a(n) for n = 1..1000

Cf. A071086, A146567, A146566, A146564.

a(n) = sum(k=1, n^2+1, if (k!=n, denominator((n*k)^2/(k^2 - n^2))==1)); \\ Michel Marcus, Oct 28 2022

More terms from Micah Manary, Aug 07 2022

cp's OEIS Frontend

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

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