Are there Any More Unitary Perfect Numbers?
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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): $87,360$