Simplification of the notion of "neutral numbers".
In the past several weeks I have been examining record-setters of the counting functions of various entities. We have several counting functions, including
A000005 (divisor counting function) →
A002182 highly composite numbers
A000010 (Euler totient function / totative counting function) (→ A000040 the prime numbers)
A010846 (regular counting function) →
A244052 highly regular numbers
A045763 ("neutral" counting function: enumerates nondivisors in the cototient of n) →
A300859 "highly neutral numbers"
A243822 (semidivisor counting function) →
A293555 "highly semidivisible numbers"
A243823 (semitotative counting function) →
A292867 "highly semitotient numbers"
In the course of doing this I have found a perhaps easier way of looking at neutral numbers.
Let
n be a nonzero positive integer; we are concerned with the range
R of integers 1 ≤
m ≤
n. We know within this range we have divisors
d |
n (see row
n of
A027750) and "totatives"
t such that gcd(
t,
n) = 1 that appear in the reduced residue system (RRS) of
n (see row
n of
A038566). Numbers
m in
R that are not coprime to
n are said to occupy the "cototient" of
n, while it is plain that numbers
m in R that do not divide
n are called "nondivisors" of
n. Therefore we can construct a set of numbers
m that are nondivisors in the cototient of
n (see row
n of
A133995), which we will deem "neutral" to
n or "the neutrals of
n". These numbers have at least one prime divisor
p that also divides
n. We know that
d |
n^
e with 0 ≤
e, an integer. We can find nondivisor
m within
R that also divide
n^e with
e > 1. Therefore we have two species of "regular"
m, divisors
d in A027750, and nondivisors
m in row
n of
A162306 that also appear in A133995; these numbers, called "semidivisors", appear in row
n of
A272618. There are numbers
m left in row
n of A133995 that do not divide any integer power of
n; these are "semitotatives" that appear in row n of
A272619.
Therefore it is simple.
We have two species of nondivisors
m in the cototient of composite
n. The first are "semidivisors"
m |
n^
e with
e > 1 for
n with more than 1 distinct prime divisor, and the last are "semitotatives"
m that do not divide any integer power of
n for composite
n > 6. We can conflate the divisors and semidivisors as "regulars"
m |
n^
e with
e ≥ 0, and semitotatives and totatives as nondivisors
m that do not divide any integer power of
n.
Interestingly, A244052 (highly regular numbers) have primorials A002110 as a subset, along with integer multiples
k < nextprime(
p) of the primorials
p# A060735, and not much correlation with highly composite A002182. The other terms that might be in A244052 appear in the "turbulent candidates" A288813, thus making A288784 the "necessary but insufficient condition" for A244052. Many of the terms in A293555 appear in A244052 and vice versa. Likewise regarding A300859 (highly neutral numbers), A293555 appears to be a subset, A244052 very closely correlated, and have a strong connection with A292867 (highly semitotient numbers). A002110 is a near-subset (all terms except the prime primorial 2). Check out my "pre-press" analysis of A300859
here, and a "concordance" of record-setters of these counting functions
here. (The studies involved a dataset for A010846 the regular counting function for all integers 1 ≤
k ≤ 36,000,000, but A300859 can be computed in reasonable time to A002110(9) = 223,092,870.)
Once I "close the circle" on the sequences related to the several secondary entities within
R, I plan to resume coding the
Tour des Bases for its re-launch in its own website, to which we might then refer from this forum if it suits us.