Sublime number

In number theory, a sublime number is a positive integer which has a perfect number of positive factors (including itself), and whose positive factors add up to another perfect number.[1]

The number 12, for example, is a sublime number. It has a perfect number of positive factors (6): 1, 2, 3, 4, 6, and 12, and the sum of these is again a perfect number: 1 + 2 + 3 + 4 + 6 + 12 = 28.

There are only two known sublime numbers, 12 and (2¹²⁶)(2⁶¹ − 1)(2³¹ − 1)(2¹⁹ − 1)(2⁷ − 1)(2⁵ − 1)(2³ − 1) (sequence A081357 in the OEIS).[2] The second of these has 76 decimal digits:

6086555670238378989670371734243169622657830773351885970528324860512791691264.

References

1. MathPages article, "Sublime Numbers".
2. Clifford A. Pickover, Wonders of Numbers, Adventures in Mathematics, Mind and Meaning New York: Oxford University Press (2003): 215