Concordance across sequences relating to nondivisors in the cototient of n.

A002110, A002182, A244052, A293555, A300156, A292867, A300860, and A300859.
Michael De Vlieger, St. Louis, Missouri, 201803161545.

Concerns sequences:
A000005: Divisor counting function τ(n), i.e., number of numbers 1 ≤ d ≤ n such that d | n^e with 0 ≤ e ≤ 1.
A000010: Euler totient function φ(n), i.e., number of 1 ≤ t < n such that gcd(t, n) = 1.
A001221: ω(n) = Number of distinct prime divisors of n.
A002110: The primorials.
A002182: Highly composite numbers, i.e., where records are set in A000005.
A010846: Regular counting function, abbreviated rcf(n), i.e., number of 1 ≤ m ≤ n such that m | n^e with e ≥ 0.
A045763: Number of nondivisors 1 < k < n such that gcd(k, n) > 1.
A243822: Number of nondivisors k in the cototient of n that divide n^e with e > 1.
A243823: Number of nondivisors k in the cototient of n that do not divide n^e.
A244052: Highly regular numbers, i.e., where records are set in A010846.
A287352 First differences of indices of prime divisors p of n (used as PC(n) = compact multiplicity notation).
A292867 Indices of records in A243823. (semitotatives).
A293555 Indices of records in A243822. (“Highly semidivisible numbers”)
A300156 Indices of records in A299990(n) = A243822(n) − A000005(n). (“Highly semidivisor-dominant numbers”)
A300858 a(n) = A243823(n) − A243822(n).
A300859 Indices of records in A045763.
A300860 Indices of records in A300858.
See [1] for these and other A-number sequences in this work.

Foreword.

Consider the range of integers 1 ≤ m ≤ n. Within this range are divisors d | n and the reduced residue system that contains the totatives t such that gcd(t, n) = 1. The set of divisors and the RRS have a common element in the empty product, 1. The numbers that are not in the RRS are said to occupy the cototient of n. The divisors d > 1 appear within the cototient for all n > 1, but for composite n > 4 there is at least one nondivisor k in the cototient of n. These k have at least one prime divisor p that also divides n.

There are two species of k; k | n^e with e > 1, and k that do not divide n^e. The former we call semidivisors, while the latter are called semitotatives of n. For composite n with ω(n) = 1, we have only semitotatives, but for n = 6, we have only one semidivisor (4) in the cototient. All other composite n have at least one of both species of k. From this, we can classify each 1 < m ≤ n as a divisor, semidivisor, semitotative, or totative with respect to n, with 1 being a totative that divides n.

We can produce record transforms of the counting functions of the four species, divisor, semidivisor, semitotative, and totative. These functions are the divisor counting function (A000005), A243822, A243823, and the Euler totient function (A000010) respectively. Further, counting functions that conflate two of the species do exist: the regular counting function A010846 = A000005 + A243822 (numbers 1 ≤ m ≤ n such that m | n^e with e ≥ 0.) and the "neutral" counting function A045763 = A243822 + A243823 = n − (A000005 + A000010 − 1). These six functions generate the following sequences that list record-setters: A002182, A293555, A292867, A006093, A244052, and A300859, in order of their presentation herein.

Regarding the conflated counting functions A010846 (regulars) and A045763 (neutrals), we produce functions that yield the difference of the component species: A299990(n) = A243822(n) − A000005(n), and A300858(n) = A243823(n) − A243822(n). Then we can produce record-setter lists for those comparisons, A300156 (semidivisor vs. divisor) and A300860 (semitotative vs. semidivisor).

This concordance produces a set T = union of all terms less than “lim” that appear in A002182, A293555, A292867, A244052, A300860, A300859, and A300156. (The sequence A006093 is too large for a concordance, but positions of the elements t of T in that sequence are furnished). This study examines the record-setting sequences for any correlation and decomposes the terms in T. We also examine the index of t in A002110 (the primorials) as the sequence is a subset of A244052.

The variable lim = 36 × 10⁶. This is the size of the A010846 dataset personally maintained.

Table 1. Concordance.

n = index
m = Union of terms < lim in A002110, A002182, A244052, A293555, A300156, A292867, A300860, and A300859.
(1) index of n in A002182, indices of records in A000005 (i.e., the highly composite numbers).
(2) index of m in A002110, the primorials, products of the smallest k primes.
(3) index of m in A244052, indices of records in A010846. (regular numbers).
(4) index of m in A293555, indices of records in A243822. (semidivisors).
(5) index of m in A300156, indices of records in A299990(n) = A243822(n) − A000005(n).
(6) index of m in A292867, indices of records in A243823. (semitotatives).
(7) index of m in A300860, indices of records in A300858(n) = A243823(n) − A243822(n).
(8) index of m in A300859, indices of records in A045763. (neutrals).
(9) index of m in A006093, indices of records in A000010. (totatives).
A287352(m) = “π-code”.

n	m	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	PC(m)
1	1	1	1	1	1	1	1	1	1	1	0
2	2	2	2	2						2	1
3	4	3		3						3	1
4	6	4	3	4	2				2	4	1.1
5	8						2	2			1.0.0
6	10			5	3				3	5	1.2
7	12	5		6						6	1.0.1
8	14						3		4		1.3
9	15							3			2.1
10	16						4	4		7	1.0.0.0
11	18			7	4				5	8	1.1.0
12	20						5				1.0.2
13	22						6		6	9	1.4
14	24	6		8							1.0.0.1
15	26						7	5	7		1.5
16	27							6			2.0.0
17	28						8	7		10	1.0.3
18	30		4	9	5	2			8	11	1.1.1
19	32						9	8			1.0.0.0.0
20	36	7							9	12	1.0.1.0
21	38						10		10		1.7
22	40						11			13	1.0.0.2
23	42			10	6	3			11	14	1.1.2
24	44						12	9			1.0.4
25	46						13			15	1.8
26	48	8					14				1.0.0.0.1
27	50						15		12		1.2.0
28	52						16	10		16	1.0.5
29	54						17		13		1.1.0.0
30	56						18	11			1.0.0.3
31	58						19			17	1.9
32	60	9		11	7				14	18	1.0.1.1
33	62						20	12			1.1
34	64						21	13			1.0.0.0.0.0
35	66					4			15	19	1.1.3
36	68						22				1.0.6
37	72						23			21	1.0.0.1.0
38	76							14			1.0.7
39	78				8	5	24		16	22	1.1.4
40	80						25	15			1.0.0.0.2
41	84			12	9				17		1.0.1.2
42	86						26				1.13
43	88						27	16		24	1.0.0.4
44	90			13	10	6			18		1.1.0.1
45	92						28				1.0.8
46	94						29				1.14
47	96						30	17		25	1.0.0.0.0.1
48	100						31	18		26	1.0.2.0
49	102					7			19	27	1.1.5
50	104							19			1.0.0.5
51	108						32			29	1.0.1.0.0
52	112							20		30	1.0.0.0.3
53	114					8	33		20		1.1.6
54	120	10		14					21		1.0.0.1.1
55	122						34	21			1.17
56	124						35	22			1.0.10
57	126				11		36		22	31	1.1.0.2
58	128							23			1.0.0.0.0.0.0
59	130						37			32	1.2.3
60	132						38		23		1.0.1.3
61	138					9	39		24	34	1.1.7
62	144						40	24			1.0.0.0.1.0
63	150			15	12	10			25	36	1.1.1.0
64	156						41			37	1.0.1.4
65	160						42	25			1.0.0.0.0.2
66	162						43			38	1.1.0.0.0
67	168								26		1.0.0.1.2
68	174						44		27		1.1.8
69	176							26			1.0.0.0.4
70	180	11		16					28	42	1.0.1.0.1
71	184							27			1.0.0.8
72	186						45		29		1.1.9
73	192						46	28		44	1.0.0.0.0.0.1
74	198								30	46	1.1.0.3
75	200							29			1.0.0.2.0
76	204						47		31		1.0.1.5
77	210		5	17	13	11			32	47	1.1.1.1
78	216						48	30			1.0.0.1.0.0
79	222						49			48	1.1.10
80	228						50			50	1.0.1.6
81	234						51		33		1.1.0.4
82	240	12					52		34	53	1.0.0.0.1.1
83	246						53	31	35		1.1.11
84	248							32			1.0.0.10
85	250							33		54	1.2.0.0
86	252						54		36		1.0.1.0.2
87	256							34		55	1.0.0.0.0.0.0.0
88	258						55		37		1.1.12
89	264						56		38		1.0.0.1.3
90	270						57		39	58	1.1.0.0.1
91	272							35			1.0.0.0.6
92	276						58	36		59	1.0.1.7
93	282						59	37		61	1.1.13
94	288						60	38			1.0.0.0.0.1.0
95	294								40		1.1.2.0
96	300						61		41		1.0.1.1.0
97	306						62			63	1.1.0.5
98	312						63			65	1.0.0.1.4
99	318						64	39	42		1.1.14
100	320							40			1.0.0.0.0.0.2

(Click here to refer to key)

n	m	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	PC(m)
----	----
101	324						65	41			1.0.1.0.0.0
102	330			18	14	12			43	67	1.1.1.2
103	336						66			68	1.0.0.0.1.2
104	342						67				1.1.0.6
105	348						68	42		70	1.0.1.8
106	354						69	43			1.1.15
107	360	13					70		44		1.0.0.1.0.1
108	366						71	44		73	1.1.16
109	372						72	45		74	1.0.1.9
110	378						73		45	75	1.1.0.0.2
111	384						74	46			1.0.0.0.0.0.0.1
112	390			19	15	13			46		1.1.1.3
113	396						75			78	1.0.1.0.3
114	402						76				1.1.17
115	408						77			80	1.0.0.1.5
116	414						78	47			1.1.0.7
117	420			20	16				47	82	1.0.1.1.1
118	426						79	48			1.1.18
119	432						80	49		84	1.0.0.0.1.0.0
120	438						81			85	1.1.19
121	444						82				1.0.1.10
122	450						83		48		1.1.0.1.0
123	456						84			88	1.0.0.1.6
124	462								49	90	1.1.2.1
125	468						85	50			1.0.1.0.4
126	474						86	51			1.1.20
127	480						87		50		1.0.0.0.0.1.1
128	486						88	52		93	1.1.0.0.0.0
129	498						89			95	1.1.21
130	504						90		51		1.0.0.1.0.2
131	510					14			52		1.1.1.4
132	516						91	53			1.0.1.12
133	522						92	54		99	1.1.0.8
134	528						93	55			1.0.0.0.1.3
135	534						94	56			1.1.22
136	540						95		53	100	1.0.1.0.0.1
137	546								54	101	1.1.2.2
138	552						96	57			1.0.0.1.7
139	558						97	58			1.1.0.9
140	564						98	59			1.0.1.13
141	570					15			55	105	1.1.1.5
142	576						99	60		106	1.0.0.0.0.0.1.0
143	582						100				1.1.23
144	588						101				1.0.1.2.0
145	594						102				1.1.0.0.3
146	600						103		56	110	1.0.0.1.1.0
147	612							61		112	1.0.1.0.5
148	618						104	62		114	1.1.25
149	624						105	63			1.0.0.0.1.4
150	630			21	17	16			57	115	1.1.0.1.1
151	636						106	64			1.0.1.14
152	642						107	65		117	1.1.26
153	648						108	66			1.0.0.1.0.0.0
154	654						109				1.1.27
155	660								58	121	1.0.1.1.2
156	666						110				1.1.0.10
157	672						111			122	1.0.0.0.0.1.2
158	684						112	67			1.0.1.0.6
159	690						113		59	125	1.1.1.6
160	696						114	68			1.0.0.1.8
161	702						115				1.1.0.0.4
162	708						116	69		127	1.0.1.15
163	714						117		60		1.1.2.3
164	720	14					118	70			1.0.0.0.1.0.1
165	732							71		130	1.0.1.16
166	738							72		131	1.1.0.11
167	744						119	73			1.0.0.1.9
168	750						120		61	133	1.1.1.0.0
169	756							74		134	1.0.1.0.0.2
170	762							75			1.1.29
171	768						121	76		136	1.0.0.0.0.0.0.0.1
172	780						122		62		1.0.1.1.3
173	786						123			138	1.1.30
174	792						124				1.0.0.1.0.3
175	798								63		1.1.2.4
176	804						125	77			1.0.1.17
177	810						126		64	141	1.1.0.0.0.1
178	816							78			1.0.0.0.1.5
179	822							79		143	1.1.31
180	828							80		145	1.0.1.0.7
181	834						127	81			1.1.32
182	840	15		22	18		128		65		1.0.0.1.1.1
183	846						129	82			1.1.0.13
184	852						130	83		147	1.0.1.18
185	858						131			149	1.1.3.1
186	864						132	84			1.0.0.0.0.1.0.0
187	870					17	133		66		1.1.1.7
188	882						134			153	1.1.0.2.0
189	900						135		67		1.0.1.0.1.0
190	906							85		155	1.1.34
191	912							86			1.0.0.0.1.6
192	924								68		1.0.1.2.1
193	930						136		69		1.1.1.8
194	936							87		159	1.0.0.1.0.4
195	948							88			1.0.1.20
196	954						137	89			1.1.0.14
197	960						138	90	70		1.0.0.0.0.0.1.1
198	966								71	163	1.1.2.5
199	972							91			1.0.1.0.0.0.0
200	990				19	18	139		72	167	1.1.0.1.2

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[1]: Neil Sloane, The Online Encyclopedia of Integer Sequences, retrieved 2018 0329 2330 GMT. See “Concerns Sequences” for individual links.

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(Updated 2 April 2018)

Concordance across sequences relating to nondivisors in the cototient of n.

Contents:

Foreword.

Table 1. Concordance.