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# Primorial numeral system

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

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

where

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

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

 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 ${\displaystyle (p_{k}-1)\cdot p_{k-1}\#,\,k\geq 1}$, rows (zero-indexed from ${\displaystyle p_{k-1}\#}$ to ${\displaystyle p_{k}\#-1}$) with ${\displaystyle k}$ 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 ${\displaystyle k=0}$.)

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, ...}

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 ${\displaystyle \phi (210)=48}$ 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 ${\displaystyle \pi (210)=46}$ 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

1. Inductive proof:

Base case:

True for ${\displaystyle n=0}$.

Induction step:

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

if and only if

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

if and only if

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