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Sublime number - Wikipedia, the free encyclopedia

Sublime number

From Wikipedia, the free encyclopedia

Divisibility-based
sets of integers
Form of factorization:
Prime number
Composite number
Powerful number
Square-free number
Achilles number
Constrained divisor sums:
Perfect number
Almost perfect number
Quasiperfect number
Multiply perfect number
Hyperperfect number
Superperfect number
Unitary perfect number
Semiperfect number
Primitive semiperfect number
Practical number
Numbers with many divisors:
Abundant number
Highly abundant number
Superabundant number
Colossally abundant number
Highly composite number
Superior highly composite number
Other:
Deficient number
Weird number
Amicable number
Friendly number
Sociable number
Solitary number
Sublime number
Harmonic divisor number
Frugal number
Equidigital number
Extravagant number
See also:
Divisor function
Divisor
Prime factor
Factorization
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In mathematics, a sublime number is a positive integer which has a perfect number of positive divisors (including itself), and whose positive divisors add up to another perfect number.[1]

The number 12, for example, is a sublime number. It has a perfect number of positive divisors (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 OEIS).[2] The second of these has 76 decimal digits:

6086555670238378 989670371734243169622657830773 351885970528324860512791691264.

[edit] 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


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