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

MathPages article, "Sublime Numbers".

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

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Divisibility-based sets of integers
Overview

Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic

Divisibility of 60
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 Bi-unitary multiply perfect Semiperfect Practical Erdős–Nicolas

With many divisors

Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird

Aliquot sequence-related

Untouchable Amicable Sociable Betrothed

Base-dependent

Equidigital Extravagant Frugal Harshad Polydivisible Smith

Other sets

Arithmetic Deficient Friendly Solitary Sublime Harmonic divisor Descartes Refactorable Superperfect

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Classes of natural numbers
Powers and related numbers

Achilles Power of 2 Power of 3 Power of 10 Square Cube Fourth power Fifth power Sixth power Seventh power Eighth power Perfect power Powerful Prime power

Of the form a × 2b ± 1

Cullen Double Mersenne Fermat Mersenne Proth Thabit Woodall

Other polynomial numbers

Carol Hilbert Idoneal Kynea Leyland Loeschian Lucky numbers of Euler

Recursively defined numbers

Fibonacci Jacobsthal Leonardo Lucas Padovan Pell Perrin

Possessing a specific set of other numbers

Knödel Riesel Sierpiński

Expressible via specific sums

Nonhypotenuse Polite Practical Primary pseudoperfect Ulam Wolstenholme

Figurate numbers
2-dimensional
centered

Centered triangular Centered square Centered pentagonal Centered hexagonal Centered heptagonal Centered octagonal Centered nonagonal Centered decagonal Star

non-centered

Triangular Square Square triangular Pentagonal Hexagonal Heptagonal Octagonal Nonagonal Decagonal Dodecagonal

3-dimensional
centered

Centered tetrahedral Centered cube Centered octahedral Centered dodecahedral Centered icosahedral

non-centered

Tetrahedral Cubic Octahedral Dodecahedral Icosahedral Stella octangula

pyramidal

Square pyramidal Pentagonal pyramidal Hexagonal pyramidal Heptagonal pyramidal

4-dimensional
non-centered

Pentatope Squared triangular Tesseractic

Combinatorial numbers

Bell Cake Catalan Dedekind Delannoy Euler Fuss–Catalan Lazy caterer's sequence Lobb Motzkin Narayana Ordered Bell Schröder Schröder–Hipparchus

Primes

Wieferich Wall–Sun–Sun Wolstenholme prime Wilson

Pseudoprimes

Carmichael number Catalan pseudoprime Elliptic pseudoprime Euler pseudoprime Euler–Jacobi pseudoprime Fermat pseudoprime Frobenius pseudoprime Lucas pseudoprime Lucas–Carmichael number Somer–Lucas pseudoprime Strong pseudoprime

Arithmetic functions and dynamics
Divisor functions

Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient Descartes Hemiperfect Highly abundant Highly composite Hyperperfect Multiply perfect Perfect Practical Primitive abundant Quasiperfect Refactorable Semiperfect Sublime Superabundant Superior highly composite Superperfect

Prime omega functions

Almost prime Semiprime

Euler's totient function

Highly cototient Highly totient Noncototient Nontotient Perfect totient Sparsely totient

Aliquot sequences

Amicable Perfect Sociable Untouchable

Primorial

Euclid Fortunate

Other prime factor or divisor related numbers

Blum Erdős–Nicolas Erdős–Woods Friendly Giuga Harmonic divisor Lucas–Carmichael Pronic Regular Rough Smooth Sphenic Størmer Super-Poulet Zeisel

Numeral system-dependent numbers
Arithmetic functions and dynamics

Persistence
Additive Multiplicative