Sublime number
# 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.

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 $(2^{126})(2^{61}$ − $1)(2^{31}$ − $1)(2^{19}$ − $1)(2^7$ − $1)(2^5$ − $1)(2^3$ − 1) (sequence A081357 in the OEIS). The second of these has 76 decimal digits:

    6,086,555,670,238,378,989,670,371,734,243,169,622,657,830,773,351,885,970,528,324,860,512,791,691,264.
