Details

The original problem [1]

may be rewritten as

,

and further rewritten as

.

Solving for leads to a complicated expression equivalent to the elliptic curve:

.

A tabulation of the integers that can be represented is given in [2].

References

[1] A. Bremner, R. K. Guy and R. J. Nowakowski, "Which Integers Are Representable as the Product of the Sum of Three Integers with the Sum of Their Reciprocals?," Mathematics of Computation, 61(203), 1993 pp. 117–130. www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1189516-5/S0025-5718-1993-1189516-5.pdf.

[2] H. Pfoertner. "Integers n representable as the product of the sum of three nonzero integers with the sum of their reciprocals." The On-Line Encyclopedia of Integer Sequences. (Jan 24, 2023) oeis.org/A085514.