Primorial numeral system

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Since we have (it is easily proved by [mathematical] induction)^[1]

\sum _{i=0}^{n}(p_{i+1}-1)\cdot p_{i}\#=p_{n+1}\#-1,\quad n\geq 0,\,

where

p_{i}\#:=\prod _{j=1}^{i}p_{j},\quad i\geq 0,\,

is the $\scriptstyle i\,$ th primorial number and $\scriptstyle p_{j}\,$ is the $\scriptstyle j\,$ th prime, and where $\scriptstyle p_{0}\#\,$ gives the empty product (defined as 1), this provides a unique representation for each natural number, with the restriction (0 to $\scriptstyle p_{i+1}-1\,$ ) on the "digits" used with the place-value $\scriptstyle p_{i}\#\,$ . This gives the primorial numeral system, also called primoradic, a mixed radix numeral system.

**Primorial numeral system (successive prime numbers as radix, whose place values are primorial numbers)**
index:	7	6	5	4	3	2	1	0
radix:	17	13	11	7	5	3	2	1
place value:	17# = 510510	13# = 30030	11# = 2310	7# = 210	5# = 30	3# = 6	2# = 2	= 1
digit:	0 to 18	0 to 16	0 to 12	0 to 10	0 to 6	0 to 4	0 to 2	0 to 1

A049345 Integers written in primorial base. (Above 10*7# - 1, we need to use "digits" separators, e.g. colons.)

{0, 1, 10, 11, 20, 21, 100, 101, 110, 111, 120, 121, 200, 201, 210, 211, 220, 221, 300, 301, 310, 311, 320, 321, 400, 401, 410, 411, 420, 421, 1000, 1001, 1010, 1011, 1020, ...}

The concatenation of

{{0}, {1}, {1, 0}, {1, 1}, {2, 0}, {2, 1}, {1, 0, 0}, {1, 0, 1}, {1, 1, 0}, {1, 1, 1}, {1, 2, 0}, {1, 2, 1}, {2, 0, 0}, {2, 0, 1}, {2, 1, 0}, {2, 1, 1}, {2, 2, 0}, {2, 2, 1}, ...}

where we have $(p_{k}-1)\cdot p_{k-1}\#,\,k\geq 1$ , rows (zero-indexed from $p_{k-1}\#$ to $p_{k}\#-1$ ) with $k$ elements gives the following sequence. (If we applied the rule of not displaying the leading 0 to zero, we would have nothing to show for it: there would be no row for $k=0$ .)

A235168 Triangle read by rows: row n gives digits of n in primorial base.

{0, 1, 1, 0, 1, 1, 2, 0, 2, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 2, 0, 1, 2, 1, 2, 0, 0, 2, 0, 1, 2, 1, 0, 2, 1, 1, 2, 2, 0, 2, 2, 1, 3, 0, 0, 3, 0, 1, 3, 1, 0, 3, 1, 1, ...}

Table of primoradics

In the following table, the primes, except for {2, 3, 5}, all congruent to {1, 7, 11, 13, 17, 19, 23, 29} (mod 30), are shown in bold italic.

In primoradic representation, except for {1:0_#, 1:1_#, 2:1_#}, all primes end with

0:0:1_#, 1:0:1_#, 1:2:1_#, 2:0:1_#, 2:2:1_#, 3:0:1_#, 3:2:1_#, 4:2:1_#

where the primes congruent to 1 (mod 6) end with

0:0:1_#, 1:0:1_#, 2:0:1_#, 3:0:1_#,

while the primes congruent to 5 (mod 6) end with

1:2:1_#, 2:2:1_#, 3:2:1_#, 4:2:1_#.

Among the $\phi (210)=48$ numbers coprime to 210 (as given by the totient function), 1 is a unit and the following five numbers

11^2 = 121, 11*13 = 143, 11*17 = 187, 11*19 = 209,

13^2 = 169,

are composite, so we have 42 primes which are coprime to 210. If we count the primes {2, 3, 5, 7} we get $\pi (210)=46$ primes up to 210 (as given by the prime counting function).

Note that starting from 10 * 7# = 2100₁₀ = 10:0:0:0:0_#, the need to use colon separators becomes obvious!

Primoradic representation of nonnegative integers from 0 to 7# − 1

n₁₀	n_#
0	0
1	1
2	1:0
3	1:1
4	2:0
5	2:1
6	1:0:0
7	1:0:1
8	1:1:0
9	1:1:1
10	1:2:0
11	1:2:1
12	2:0:0
13	2:0:1
14	2:1:0
15	2:1:1
16	2:2:0
17	2:2:1
18	3:0:0
19	3:0:1
20	3:1:0
21	3:1:1
22	3:2:0
23	3:2:1
24	4:0:0
25	4:0:1
26	4:1:0
27	4:1:1
28	4:2:0
29	4:2:1

n₁₀	n_#
30	1:0:0:0
31	1:0:0:1
32	1:0:1:0
33	1:0:1:1
34	1:0:2:0
35	1:0:2:1
36	1:1:0:0
37	1:1:0:1
38	1:1:1:0
39	1:1:1:1
40	1:1:2:0
41	1:1:2:1
42	1:2:0:0
43	1:2:0:1
44	1:2:1:0
45	1:2:1:1
46	1:2:2:0
47	1:2:2:1
48	1:3:0:0
49	1:3:0:1
50	1:3:1:0
51	1:3:1:1
52	1:3:2:0
53	1:3:2:1
54	1:4:0:0
55	1:4:0:1
56	1:4:1:0
57	1:4:1:1
58	1:4:2:0
59	1:4:2:1

n₁₀	n_#
60	2:0:0:0
61	2:0:0:1
62	2:0:1:0
63	2:0:1:1
64	2:0:2:0
65	2:0:2:1
66	2:1:0:0
67	2:1:0:1
68	2:1:1:0
69	2:1:1:1
70	2:1:2:0
71	2:1:2:1
72	2:2:0:0
73	2:2:0:1
74	2:2:1:0
75	2:2:1:1
76	2:2:2:0
77	2:2:2:1
78	2:3:0:0
79	2:3:0:1
80	2:3:1:0
81	2:3:1:1
82	2:3:2:0
83	2:3:2:1
84	2:4:0:0
85	2:4:0:1
86	2:4:1:0
87	2:4:1:1
88	2:4:2:0
89	2:4:2:1

n₁₀	n_#
90	3:0:0:0
91	3:0:0:1
92	3:0:1:0
93	3:0:1:1
94	3:0:2:0
95	3:0:2:1
96	3:1:0:0
97	3:1:0:1
98	3:1:1:0
99	3:1:1:1
100	3:1:2:0
101	3:1:2:1
102	3:2:0:0
103	3:2:0:1
104	3:2:1:0
105	3:2:1:1
106	3:2:2:0
107	3:2:2:1
108	3:3:0:0
109	3:3:0:1
110	3:3:1:0
111	3:3:1:1
112	3:3:2:0
113	3:3:2:1
114	3:4:0:0
115	3:4:0:1
116	3:4:1:0
117	3:4:1:1
118	3:4:2:0
119	3:4:2:1

n₁₀	n_#
120	4:0:0:0
121	4:0:0:1
122	4:0:1:0
123	4:0:1:1
124	4:0:2:0
125	4:0:2:1
126	4:1:0:0
127	4:1:0:1
128	4:1:1:0
129	4:1:1:1
130	4:1:2:0
131	4:1:2:1
132	4:2:0:0
133	4:2:0:1
134	4:2:1:0
135	4:2:1:1
136	4:2:2:0
137	4:2:2:1
138	4:3:0:0
139	4:3:0:1
140	4:3:1:0
141	4:3:1:1
142	4:3:2:0
143	4:3:2:1
144	4:4:0:0
145	4:4:0:1
146	4:4:1:0
147	4:4:1:1
148	4:4:2:0
149	4:4:2:1

n₁₀	n_#
150	5:0:0:0
151	5:0:0:1
152	5:0:1:0
153	5:0:1:1
154	5:0:2:0
155	5:0:2:1
156	5:1:0:0
157	5:1:0:1
158	5:1:1:0
159	5:1:1:1
160	5:1:2:0
161	5:1:2:1
162	5:2:0:0
163	5:2:0:1
164	5:2:1:0
165	5:2:1:1
166	5:2:2:0
167	5:2:2:1
168	5:3:0:0
169	5:3:0:1
170	5:3:1:0
171	5:3:1:1
172	5:3:2:0
173	5:3:2:1
174	5:4:0:0
175	5:4:0:1
176	5:4:1:0
177	5:4:1:1
178	5:4:2:0
179	5:4:2:1

n₁₀	n_#
180	6:0:0:0
181	6:0:0:1
182	6:0:1:0
183	6:0:1:1
184	6:0:2:0
185	6:0:2:1
186	6:1:0:0
187	6:1:0:1
188	6:1:1:0
189	6:1:1:1
190	6:1:2:0
191	6:1:2:1
192	6:2:0:0
193	6:2:0:1
194	6:2:1:0
195	6:2:1:1
196	6:2:2:0
197	6:2:2:1
198	6:3:0:0
199	6:3:0:1
200	6:3:1:0
201	6:3:1:1
202	6:3:2:0
203	6:3:2:1
204	6:4:0:0
205	6:4:0:1
206	6:4:1:0
207	6:4:1:1
208	6:4:2:0
209	6:4:2:1

Notes

↑ Inductive proof:

Base case:

True for $n=0$ .

Induction step:

$\sum _{i=0}^{n}(p_{i+1}-1)\cdot p_{i}\#=p_{n+1}\#-1,\quad n\geq 0,\,$

if and only if

$\sum _{i=0}^{n-1}(p_{i+1}-1)\cdot p_{i}\#+(p_{n+1}-1)\cdot p_{n}\#=p_{n+1}\cdot p_{n}\#-1,\quad n\geq 0,\,$

if and only if

$\sum _{i=0}^{n-1}(p_{i+1}-1)\cdot p_{i}\#=p_{n}\#-1,\quad n\geq 0.\,$

[1] Inductive proof:

Base case:

True for $n=0$ .

Induction step:

$\sum _{i=0}^{n}(p_{i+1}-1)\cdot p_{i}\#=p_{n+1}\#-1,\quad n\geq 0,\,$

if and only if

$\sum _{i=0}^{n-1}(p_{i+1}-1)\cdot p_{i}\#+(p_{n+1}-1)\cdot p_{n}\#=p_{n+1}\cdot p_{n}\#-1,\quad n\geq 0,\,$

if and only if

$\sum _{i=0}^{n-1}(p_{i+1}-1)\cdot p_{i}\#=p_{n}\#-1,\quad n\geq 0.\,$

[1]

Primorial numeral system

Table of primoradics

See also

Notes

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