Trimorphic number

In mathematics a trimorphic number is a number whose cube (expressed in a given base) ends in the number itself. For example, 4³ = 64, 24³ = 13824, and 249³ = 15438249.

The first few trimorphic numbers in base 10 are:

0, 1, 4, 5, 6, 9, 24, 25, 49, 51, 75, 76, 99, 125, 249, 251, 375, 376, 499, 501, 624, 625, 749, 751, 875, 999, 1249, 3751, 4375, 4999, 5001, 5625, 6249, 8751, 9375, 9376, 9999, ... (sequence A033819 in the OEIS)

Every automorphic number is also a trimorphic number, but there are trimorphic numbers which are not automorphic (such as 4, 9 and 24).

Other bases

In base 12, the trimorphic numbers are:

0, 1, 3, 4, 5, 7, 8, 9, Ɛ, 15, 47, 53, 54, 5Ɛ, 61, 68, 69, 75, ᘔ7, Ɛ3, ƐƐ, 115, 253, 368, 369, 4ᘔ7, 5ƐƐ, 601, 715, 853, 854, 969, ᘔᘔ7, ƐƐƐ, 14ᘔ7, 2369, 3853, 3854, 4715, 5ƐƐƐ, 6001, 74ᘔ7, 8368, 8369, 9853, ᘔ715, ƐƐƐƐ, ...

References

Weisstein, Eric W. "Trimorphic Number". MathWorld.

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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