**Neutral Digits.**
I just wrote a paper on neutral digits. I'll post a link here once it's been reviewed. This is not a "canonical" elementary number theory topic, however the definitions and conjectures are proved in the paper.

Let the integer

*r* >= 2 be a number base. Set d(

*r*) as the number of divisors and t(

*r*) as the number of totatives (I am using t in place of the Greek letter phi, since there is some problem displaying phi right now).

We can produce a "neutral digit counting function" using the following formula:

n(

*r*) = d(

*r*) + t(

*r*) - 1

Note, this formula appears at the OEIS sequence

A045763, which is by this same definition, the set of neutral digits. (Edited 12 June 2014.)

We subtract 1 because it is counted both in d(

*r*) and t(

*r*), since it is a divisor of and coprime to

*r*. Observe that many of the composite numbers

*r* > 4 have a significant difference between the number of divisors and non-unitary totatives:

(in the case of

*r* = 2, the difference is -1 since there is only one totative, digit-1.)

In the paper I prove that there are neutral digits for every composite

*r* > 4, and that there are two possible kinds of neutral digit. I've called the two kinds of neutral digits the "semidivisor" and the "semitotative".

**Semidivisor.**
Let the integer

*p* be a prime divisor of

*r*. A semidivisor is a regular neutral digit that does not divide

*r* evenly, and is the product of at least two prime divisors

*p*.

To define the semidivisor we need the

fundamental theorem of arithmetic, and the unique factorization / prime power decomposition formula. See sources [A] page 86, [ B] pages 16-18, [E] page 3, [F] page 20. Let the integer

*k* >= 1, and let the integer

*p* be a distinct prime divisor of

*r*. Let the integer

*e* > 0 be the multiplicity (i.e., the exponent) of

*p* in

*r*.

Then a divisor

*d* of

*r* can be expressed as

It is too "rich" in at least one prime divisor

*p*, when compared to the base

*r*. Digit 4 base 6 has a prime divisor 2 with multiplicity 2, whilst there is only one 2 in 6. Digit 8 base 10 has a prime divisor 2 with multiplicity 3, whilst there is only one 2 in 10.

I wrote an algorithm in autumn 2013 that can produce the quantity of semidivisors for any given base. This was submitted to the OEIS 11 June 2014 and is now sequence

A243822. The semidivisor also appears in an article I wrote for the ACM Inroads in early 2012. Note that a sequence that quantifies regular digits for bases

*n* appears at

A010846 (Edited 12 June 2014.)

The semidivisor, like other regular numbers, is generally an asset for those interested in human intuitive computation. In the multiplication table the semidivisor shares the shorter end-digit periods as divisors (there is a relationship between the semidivisor and a complement in these cycles). A semidivisor enjoys a terminating digital expression of its reciprocal. One eighth in decimal is 0.125. One dozen-fourth, a regular number in dozenal is 0;09. The number of places after the radix point will be greater than one, related to the "maximum multiplicity differential" of the semidivisor and the number base. Calculate the maximum multiplicity differential by finding which prime divisor has the highest multiplicity difference with the corresponding prime in the number base. The regular divisibility tests apply to semidivisors. This is a stumbling block for "remote" or "enriched" semidivisors, i.e., those having a high maximum multiplicity differential. Though one can use the regular divisibility test in decimal for eight, this involves knowing the 125 combinations of the three end digits divisible by eight {000, 008, 016, 024, 032, 040, 048, ... 984, 992}.

I've proved in the paper that all composite bases

*r* that are not powers of primes have at least one semidivisor. This rules out base 4 for neutral digits, as it is the square of 2. Base 6 is the smallest base that has a semidivisor, digit-4.

**Semitotative.**
The semitotative is easier to define. Let the integer

*q* be a prime that is coprime to

*r*. A semitotative is a non-regular neutral digit that is the product of at least one prime divisor

*p* and at least one prime totative

*q*.

In the paper I proved that a given number base, an integer

*r* >= 2 will have at least one semitotative if a minimum totative

*q* is smaller than the complement to the minimum prime divisor

*p*. Thus, every composite number

*r* > 6 will have semitotatives. This leaves

*r* = 4 without neutral digits, and makes

*r* = 6 unique in that it possesses semidivisors, but no semitotatives.

I wrote an algorithm in autumn 2013 that can produce the quantity of semitotatives for any given base. This was submitted to the OEIS 11 June 2014 and is now sequence

A243823. The semitotative also appears in an article I wrote for the ACM Inroads in early 2012. (Edited 12 June 2014.)

The semitotative behaves like a semidivisor in the multiplication table, with short end-digit patterns. The digital expansion of multiples of reciprocals of semitotatives is recurrent. Semitotatives as digital reciprocals feature a brief, non-repeating set of digits, then a recurrent mantissa. One sixth in base ten is 0.1666..., one tenth in base twelve is 0;124972497... See source [E] page 142. Like the totative, the semitotative possesses no intuitive divisibility rule unless it is a product of two coprime factors that possess regular or neighbor-related intuitive divisibility tests. This is what I call a "compound intuitive divisibility test" in an upcoming paper.

Thus, with the neutral digits, we have a full "spectrum" of digits with direct relationships to the number base

*r*. This enables the production of digit maps, which indicate the "usefulness" of a number base

*r*, usefulness defined as being beneficial to the intuitive human computation in base

*r*, ignoring the effects of magnitude of

*r*. We can use "digit maps" of number bases

*r* as a shorthand for their utility, since the types of digits speak to their behavior in a few basic applications (multiplication tables, divisibility tests, digital fractions, etc.)