Highly totient number

A highly totient number k is an integer that has more solutions to the equation φ(x) = k, where φ is Euler's totient function, than any integer below it. The first few highly totient numbers are

1, 2, 4, 8, 12, 24, 48, 72, 144, 240, 432, 480, 576, 720, 1152, 1440 (sequence A097942 in the OEIS), with 1, 3, 4, 5, 6, 10, 11, 17, 21, 31, 34, 37, 38, 49, 54, and 72 totient solutions respectively. The sequence of highly totient numbers is a subset of the sequence of smallest number k with exactly n solutions to φ(x) = k.^[1]

The totient of a number x, with prime factorization $x=\prod_i p_i^{e_i}$ , is the product:

\phi(x)=\prod_i (p_i-1)p_i^{e_i-1}.

Thus, a highly totient number is a number that has more ways of being expressed as a product of this form than does any smaller number.

The concept is somewhat analogous to that of highly composite numbers, and in the same way that 1 is the only odd highly composite number, it is also the only odd highly totient number (indeed, the only odd number to not be a nontotient). And just as there are infinitely many highly composite numbers, there are also infinitely many highly totient numbers, though the highly totient numbers get tougher to find the higher one goes, since calculating the totient function involves factorization into primes, something that becomes extremely difficult as the numbers get larger.

References

↑ OEIS A097942

L. Havelock, A Few Observations on Totient and Cototient Valence from PlanetMath

Totient function

Euler's totient function $\phi (n)$ Jordan's totient function $J_{k}(n)$ Carmichael function $\lambda (n)$ Nontotient Noncototient Highly totient number Highly cototient number Sparsely totient number

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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