Primorial Number System

 I've written about the factorial number system in two previous posts:

• Factorial Number Base on March 31st 2018
• Factorial Number System on September 2nd 2020

It is a mixed radix number system and so is the primorial number system that uses the primorials (progressive products of primes):

• 2
• 2 x 3 = 6
• 2 x 3 x 5 = 30
• 2 x 3 x 5 x 7 = 210
• 2 x 3 x 5 x 7 x 11 = 2310
• 2 x 3 x 5 x 7 x 11 x 13 = 30030
• 2 x 3 x 5 x 7 x 11 x 13 x 17 = 510510
• 2 x 3 x 5 x 7 x 11 x 13 x 17 x 19 = 9699690 etc.

It's easy enough to set up an algorithm in SageMath that will convert decimal number to primorial digits. In decreasing order, the primorials below ten million are 9699690, 510510, 30030, 2310, 210, 30, 6 and 2. These numbers can serve as bases to represent any number up to 10,242,789 (which has representation as 111111111). Here is the algorithm (permalink) that will return the primorial base representation of any number up to 10,242,789. The example returns the representation for numbers from 2 to 50:

P=[9699690, 510510,30030,2310,210,30,6,2]

for number in [1..50]:

    original=number

    N=[]

    for p in P:

        N.append(number//p)

        number=number%p

    if is_odd(original):

        N.append(1)

    else:

        N.append(0)

    primorial=""

    for n in N:

        primorial+=str(n)

    print(original,"-->", int(primorial))

Primorial Formula

1 --> 1

2 --> 10

3 --> 11

4 --> 20

5 --> 21

6 --> 100

7 --> 101

8 --> 110

9 --> 111

10 --> 120

11 --> 121

12 --> 200

13 --> 201

14 --> 210

15 --> 211

16 --> 220

17 --> 221

18 --> 300

19 --> 301

20 --> 310

21 --> 311

22 --> 320

23 --> 321

24 --> 400

25 --> 401

26 --> 410

27 --> 411

28 --> 420

29 --> 421

30 --> 1000

31 --> 1001

32 --> 1010

33 --> 1011

34 --> 1020

35 --> 1021

36 --> 1100

37 --> 1101

38 --> 1110

39 --> 1111

40 --> 1120

41 --> 1121

42 --> 1200

43 --> 1201

44 --> 1210

45 --> 1211

46 --> 1220

47 --> 1221

48 --> 1300

49 --> 1301

50 --> 1310

Figure 1 shows a comparison of the factorial and primorial number systems:

Figure 1: source

The source for Figure 1 contains information on a wide range of unusual number systems which I may post about at some time in the future.

So what stimulated my interest in primorial number systems? Well, like most of my posts, it was prompted by my analysis of the number representing my diurnal age. Today I turned 26250 days old and this number has a striking factorisation:$$26250=7 \times 5^4 \times 3 \times  2$$The first entry for this number in the OEIS is A276086:

A276086

Prime product form of primorial base expansion of \(n\): digits in primorial base  representation of \(n\) become the exponents of successive prime factors whose product a(\(n\)) is.

It took me some time to understand what this meant. In the case of 26250, \(n\)=57 and its primorial base expansion is 1411. The resulting digits because the exponents of successive prime factors that multiply together to give 26250. In other words:$$7^1 \times 5^4 \times 3^1 \times 2^1 = 26250$$