Are there Any More Unitary Perfect Numbers?

Open Question

Apart from the ones which are known about, are there any more unitary perfect numbers?

From Sequence of Unitary Perfect Numbers:


The sequence of unitary perfect numbers begins:

$6, 60, 90, 87 \, 360, 146 \, 361 \, 946 \, 186 \, 458 \, 562 \, 560 \, 000, \ldots$

It is not known whether this sequence is complete.


Progress

It is known that any larger unitary perfect number must have the following properties:

It must have at least $8$ prime factors
It must be divisible by a power of $10$ greater than $2^{10}$.


Sources

  • 1994: Richard K. Guy: Unsolved Problems in Number Theory (2nd ed.)
  • 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $87,360$
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