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:
summa1.png(Map for 1 ≤
n ≤ 210)
Check out
the map of all "digits" of "bases" less than 2310. Hearts ♥ attempt to key the relationships to this map.
Divisors:
A000005: Divisor counting function (
σ_0(
n),
τ(n),
d(
n)).
A027750: Divisors of
n.
A051731: Characteristic function of the divisors: 1 if
m |
n, 0 if not (useful in producing maps).
♥ These show on the map as red pixels. The first pixel is always purple
♥, as 1 is also a totative.
A002182: A000005 recordsetters. (Highly composite numbers).
A002183: Records in A000005.
Totatives:
A000010: Euler totient function (Totative counting function) (
ϕ(
n)).
A038566: Totatives of
n (reduced residue system of
n).
A054521: Characteristic function of the totatives: 1 if gcd(
m,
n) = 1, 0 if not.
♥ These show on the map as light gray pixels. The first pixel is always purple
♥, as 1 is also a divisor.
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.
♥♥♥ These show on the map as red, orange, or yellow pixels.
Nondivisors:
A049820: Nondivisor counting function.
A173540: Nondivisors of
n.
♥♥♥ These show on the map as orange, yellow, or light gray pixels.
Nondivisors in the cototient (numbers 1 <
m <
n "neutral" to
n):
A045763: Neutral counting function.
A133995: Neutrals of
n.
A304571: Characteristic function of nondivisors in the cototient of
n.***
♥♥ These show on the map as orange or yellow pixels.
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.
( rcf(n). We can calculate regulars m < lim for lim > n with rcf(n, lim) )
A162306: regulars of
n.
A304569: Characteristic function of numbers
m such that
m |
n^e with e ≥ 0.*
♥♥♥ These show on the map as purple, red, or orange pixels.
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.*
A304570: Characteristic function of numbers
m such that
m |
n^
e with
e > 1.***
♥ These show on the map as orange pixels.
A293555: A243822 recordsetters.*
A293556: Records in A243822.*
Semitotatives (nonregulars in the cototient of
n):
A243823: Semitotative counting function.*
A272619: Semitotatives of
n.*
A304572: Characteristic function of numbers
m such that
m divides no integer power of
n yet gcd(
m,
n) > 1.***
♥ These show on the map as yellow pixels.
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).*
A3xxxxx: rcf(A002110(i), m) - A010846(m) for m in A288813 (Deficit of rcf(m) versus rcf(n, m)).
"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.*
Numbers
m highly composite or superabundant, plotted as (
x,
y) = (
m/
p_
ω(
m)#,
ω(
m)).
A301413:
m/
p_
ω(
m)# for highly composite
m.
A3xxxxx:
m/
p_
ω(
m)# for superabundant
m.
A301416:
m/
p_
ω(
m)# for superior highly composite
m.
A3xxxxx:
m/
p_
ω(
m)# for colossally abundant
m.
A301414: Primitive values in A301413.
A305025: Primitive values of
m/
p_
ω(
m)# for superabundant m.
A3xxxxx: Primitive values of
m/
p_
ω(
m)# for superior highly composite
m.
A3xxxxx: Primitive values of
m/
p_
ω(
m)# for colossally abundant
m.
A301415: Number of primorials such that A301414(
n) produces highly composite
m.
A3xxxxx: Number of primorials such that primitive values of
m/
p_
ω(
m)# produces superabundant
m.
A3xxxxx: Smallest
k such that
p_
k# × A301414(
n) produces highly composite
m.
A3xxxxx: Largest
k such that
p_
k# × A301414(
n) produces highly composite
m.
A3xxxxx: Smallest
k such that
p_
k# × primitive values of
m/
p_
ω(
m)# produces superabundant
m.
A3xxxxx: Largest k such that p_k# × primitive values of
m/
p_
ω(
m)# produces superabundant
m.
* sequences I'd added based on research presented here.
** current drafts.
*** prepared sequences with reserved A-numbers.
Updated 201805231745 CDT.
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