Extravagant number

An extravagant number (also known as a wasteful number) is a natural number that has fewer digits than the number of digits in its prime factorization (including exponents).^[1] For example, in base-10 arithmetic 4 = 2², 6 = 2×3, 8 = 2³, and 9 = 3² are extravagant numbers.

Extravagant numbers can be defined in any base. There are infinitely many extravagant numbers, no matter what base is used.^[1]

Notes

1 2 Darling, David J. (2004). The universal book of mathematics: from Abracadabra to Zeno's paradoxes. John Wiley & Sons. p. 102. ISBN 978-0-471-27047-8.

References

R.G.E. Pinch (1998), Economical Numbers.
Chris Caldwell, The Prime Glossary: extravagant number at The Prime Pages.

Divisibility-based sets of integers

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

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 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

Other sets	Deficient Friendly Solitary Sublime Harmonic divisor Frugal Equidigital Extravagant

Classes of natural numbers

Powers and
related numbers

Of the form
a × 2^b ± 1

Other polynomial
numbers

Recursively defined
numbers

Possessing a
specific set
of other numbers

Expressible via
specific sums

Generated
via a sieve

Lucky

Code related

Meertens

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 Octahedral Dodecahedral Icosahedral Stella octangula

pyramidal	Square pyramidal Pentagonal pyramidal Hexagonal pyramidal Heptagonal pyramidal

4-dimensional

centered	Centered pentachoric Squared triangular

non-centered	Pentatope

Pseudoprimes

Combinatorial
numbers

Arithmetic functions

By properties of σ(n)	Abundant Almost perfect Arithmetic Colossally abundant Descartes Hemiperfect Highly abundant Highly composite Hyperperfect Multiply perfect Perfect Practical number Primitive abundant Quasiperfect Refactorable Sublime Superabundant Superior highly composite Superperfect

By properties of Ω(n)	Almost prime Semiprime

By properties of φ(n)	Highly cototient Highly totient Noncototient Nontotient Perfect totient Sparsely totient

By properties of s(n)	Amicable Betrothed Deficient Semiperfect

Dividing a quotient

Other prime factor
or divisor related
numbers

Recreational
mathematics

Base-dependent
numbers

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Extravagant number

See also

Notes

References