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.

As of July 2026, 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). 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.

It has 8128 positive divisors, which all add up to 2126(2127-1), the 12th even perfect number (by the Euclid–Euler theorem, since 2127-1 is the 12th Mersenne prime).

References

  1. ↑ MathPages article, "Sublime Numbers".
  2. 1 2 Weisstein, Eric W. "Sublime Number". Wolfram Mathworld. Wolfram Research, Inc. Retrieved 2026-10-25.
  3. ↑ Darling, David (2008-04-21). "Sublime Number". The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes. Turner Publishing Company. ISBN 978-0-470-30788-5.
  4. ↑ Clifford A. Pickover, Wonders of Numbers, Adventures in Mathematics, Mind and Meaning New York: Oxford University Press (2003): 215