Semidivisor

Let n ≥ 2 be an integer number base. A semidivisor is a necessarily composite n-regular number r | n^ε with richness ε > 1. Primes p and prime powers p^m have semidivisors that exceed the base and are also perfect powers p^k with k > m. All composite numbers n have at least one semidivisor r < n. The smallest case of semidivisor is r = 4, n = 6. Almost all n-regular numbers r for n > 1 are semidivisors.

The map below plots semidivisors in shades of orange; the shade is lighter for increasingly impractical semidivisors. Number base n appears on the vertical axis and digit k appears on the horizontal.

semidivisor map

A kind of neutral number

The semidivisor r = ξ_d is one of two kinds of neutral numbers ξ with respect to base n; i.e., these are nondivisors that are also not coprime to n, and thus are not counted by the divisor counting function τ(n) and the Euler totient function φ(n).

A kind of regular number

The divisor d is a special case of n-regular number such that d | n^ε with 0 ≤ ε ≤ 1, with d = 1 the only divisor that divides n⁰. A semidivisor is simply a nondivisor regular number r | n^ε with ε > 1.

Therefore, the semidivisor r is both regular and neutral to n.

The semidivisor counting function

The semidivisor counting function ξ_d(n) = A243822(n) is the number of semidivisors ξ_d < n:

ξ_d(n) = RCF(n) – τ(n),
= ξ(n) − ξ_t(n)

or in terms of the OEIS:

A243822(n) = A010846(n) − A000005(n),
= A045763(n) − A243823(n)

For prime p, ξ_d(p) = 0. In other words, prime bases p do not have semidivisors smaller than p.

For prime powers p^m, ξ_d(p^m)= 0. All semidivisors of prime power bases are larger than the base.
Furthermore, these semidivisors are powers p^k with k > m.

All composite n that are not prime powers have at least one semidivisor ξ_d < n.

The number n = 6 is the only n with ξ_d > 0 but ξ_t = 0. That is, with a semidivisor less than n (i.e., 4) but no semitotative.

Behavior of semidivisors in base n

Fractions expanded in base n with a semidivisor ξ_d as the denominator terminate after more than 1 digit, e.g., decimal ¼ = .25, while duodecimal ¼ = .3.
Semidivisors have a multiplicative complement g′ such that n^ε = ξ_d· g′.
Semidivisors have periods in the multiplication table λ = GCD(n, ξ_d), and unit cycles κ that are terminating rational numbers. The periods and cycles are not equal (λ ≠ κ).
Unit fractions 1/ξ_d expanded in base n involve ε places after the radix point, with the base-n representation of g′at the n ^−ε place.
We can test for divisibility of an arbitrary nonzero integer x by s_d in base r by examining ρ least significant digits: if this value matches one of the g′ multiples of ξ_d ≤ n^ε (with place value “0” signifying x ≡ 0 (mod r)) then x is divisible by ξ_d.
Base-n divisibility tests that pertain to semidivisors involve examining two or more final digits.
Only semidivisors with low richnesses may prove practical to use in base n.
Any composite number can have a compound divisibility rule involving all the divisibility tests of each distinct prime power p^μ of ξ_d. At times it may be easier to rely on these compound regular divisibility tests than the regular IDT.
The minimum case of semidivisor ξ_d is digit 4 base 6. Note that all integer bases n have semidivisors ξ_d > n. Any perfect power of n is a semidivisor.

Semidivisors in the OEIS

There are a few sequences in the OEIS that examine the semidivisors 1 < ξ_d < n:

A243822: The semidivisor counting function ξ_d(n) = the number of semidivisors ξ_d < n.
A272618: Row n of this sequence lists semidivisors ξ_d < n.
A293555: Numbers n that set records for the semidivisor counting function ξ_d(n).
A295523: Numbers n that have more semidivisors than semitotatives (the two kinds of n-neutral number).
A299991: Numbers n that have more semidivisors than divisors (two kinds of n-regular number).
A299992: Numbers n that have fewer semidivisors than divisors.
A300155: Numbers n that have equal numbers of semidivisors and divisors.
A300156: Numbers n that set records for the difference between the semidivisor counting function ξ_d(n) and the divisor counting function τ(n).
A289280: Smallest n-regular r > n (the smallest semidivisor r greater than n).

Color Canon

The colors used to represent semidivisors in this work appear in the table below:

color canon for semidivisors

Provenance of the term “semidivisor”

The term “semidivisor” is a 2008 coinage of the author of this work. The term applied to one of two species of n-neutral number which was a composite product restricted to primes p | n. In researching an extant name for such an entity, the author turned up no standard. The term “quasi-divisor” was also considered, but semidivisor became favored mainly through euphony with the term for the other n-neutral entity, the semitotative (or semicoprime), as the two terms for the species influenced one another.

The term attempts to succinctly describe an integer k that “closely approaches” dividing n, instead dividing some perfect power of n.

References

Michael De Vlieger, Exploring Number Bases as Tools, ACM Inroads, 2012, Vol. 3, No. 1, 4-12.