Sublime number

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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.

As of May 2025^[update], 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:

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.

References

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

[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

[1]

[2]

v t e Divisibility-based sets of integers
Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic
Factorization forms	Prime Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular Rough Unusual
Constrained divisor sums	Perfect Almost perfect Quasiperfect Multiply perfect Hemiperfect Hyperperfect Superperfect Unitary perfect Semiperfect Practical Descartes Erd?s–Nicolas
With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird
Aliquot sequence-related	Untouchable Amicable (Triple) Sociable Betrothed
Base-dependent	Equidigital Extravagant Frugal Harshad Polydivisible Smith
Other sets	Arithmetic Deficient Friendly Solitary Sublime Harmonic divisor Refactorable Superperfect

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