Number Base Theory 101

Dozens Demigod
Dozens Demigod
Joined: 11 Apr 2006, 12:29

27 Feb 2018, 18:35 #25

Summary of OEIS sequences associated with number theory of interest to number bases (m1n1f1g's "radicologists", or the OP's "number base theory 101") and digit maps:

A000005: Divisor counting function (σ_0(n), τ(n), d(n)).
A027750: Divisors of n.
A002182: A000005 recordsetters. (Highly composite numbers).
A002183: Records in A000005.

A000010: Euler totient function (Totative counting function) (ϕ(n)).
A038566: Totatives of n (reduced residue system of n).
A000040: A000010 recordsetters (the primes).
A006093: Records in A000010 (primes p - 1).

Cototient: (non-totatives):
A051953: Cototient counting function.
A121998: Numbers m < n in the cototient of n.
A065385: A051953 recordsetters.
A065386: Records in A051953.

A049820: Nondivisor counting function.
A173540: Nondivisors of n.

Nondivisors in the cototient (numbers 1 < m < n "neutral" to n):
A045763: Neutral counting function.
A133995: Neutrals of n.
A300859: A045763 recordsetters (highly neutral numbers)*
A300914: Records in A045763*

Regulars (here, numbers 1 < m < n such that m divides n^e with e ≥ 0):
A010846: Regular counting function.
A162306: regulars of n.
A244052: A010846 recordsetters (Highly regular numbers).*
A244053: Records in A010846.*

Semidivisors (numbers 1 < m < n such that m divides n^e with e > 1):
A243822: Semidivisor counting function.*
A272618: Semidivisors of n.*
A293555: A243822 recordsetters.*
A293556: Records in A243822.*

Semitotatives (nonregulars in the cototient of n):
A243823: Semitotative counting function.*
A272619: Semitotatives of n.*
A292867: A243823 recordsetters.*
A293868: Records in A243823.*

Richness of regulars of n: (richness meaning the least power n^e that regular m divides):
A279907: Richness of numbers in the range n.*
A280269: Richness of row n of A162306.*
A280274: Maximum richness in row n of A162306.*
A280363: Underlying formula for A280274.*
A294306: Population of values in row n of A280269.*

Study of "Highly Regular Numbers" A244052 (2016-7 "Turbulent Candidates" paper):
A288784: Necessary but insufficient condition.*
A288813: Turbulent candidates in A288784.*
A289171: "Depth"-"Distension" correlation for primorial(n).*

"Dominance" studies:
A294575: Semitotative-dominant numbers.*
A294576: Odd Semitotative-dominant numbers.*
A295221: Semitotative parity numbers.*
A295523: Nonprimes that have more semidivisors than semitotatives.*
A294492: Recordsetters for A045763(n)/n.*

Semidivisors vs. Divisors:
A299990: A243822(n) - A000005(n).*
A299991: Numbers that have more semidivisors than divisors.*
A299992: Numbers with more than 1 distinct prime divisor that have fewer semidivisors than divisors.*
A300155: Numbers that have equal numbers of semidivisors and divisors.*
A300156: A299990 recordsetters.*
A300157: Records in A299990.*

Semitotatives vs. Semidivisors:
A300858: A243823(n) - A243822(n).** (A300858(p) for p prime = 0, for n = {6, 10, 12, 18, 30}, A300858(n) is negative.)
A300860: A300858 recordsetters.**
A300861: Records in A300858.**

* sequences I'd added based on research presented here.
** current drafts.