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# LCM numeral system

The (tentatively named) LCM numeral system[1] uses the least common multiple (LCM) of the first ${\displaystyle \scriptstyle k+1}$ prime powers as place value for index ${\displaystyle k}$, where the first prime power ${\displaystyle \scriptstyle p^{0}=1}$ is the place value for index 0.

## Representation of integers

The following table shows how to proceed to get a unique representation for all nonnegative integers. (This is definitively a very unwieldy numeral system: you have to be very careful about the range of digits for nonnegative index ${\displaystyle k}$, which goes from ${\displaystyle 0}$ up to ${\displaystyle r_{k+1}-1}$, where ${\displaystyle r_{k+1}}$ is the radix for index ${\displaystyle k+1}$.)

 index: 7 6 5 4 3 2 1 0 radix: 3 2 7 5 2 3 2 1 place value: LCM(1, .., 3^2) = 2520 LCM(1, .., 2^3) = 840 LCM(1, .., 7) = 420 LCM(1, .., 5) = 60 LCM(1, .., 2^2) = 12 LCM(1, .., 3) = 6 LCM(1, .., 2) = 2 LCM(1, .., 1) = 1 max weight 10 × 2520 = 25200 2 × 840 = 1680 1 × 420 = 420 6 × 60 = 360 4 × 12 = 48 1 × 6 = 6 2 × 2 = 4 1 × 1 = 1 max value 25200 + 2519 = 27719 1680 + 839 = 2519 420 + 419 = 839 360 + 59 = 419 48 + 11 = 59 6 + 5 = 11 4 + 1 = 5 1 = 1 digit: 0 to 10 0 to 2 0 to 1 0 to 6 0 to 4 0 to 1 0 to 2 0 to 1

A025473 a(1) = 1; for n > 1, a(n) = prime root of n-th prime power (A000961).
(LCM numeral system: A025473(n+1) is radix for index n, n ≥ 0; A025473(-n+1) is 1/radix for index n, n < 0.)

{1, 2, 3, 2, 5, 7, 2, 3, 11, 13, 2, 17, 19, 23, 5, 3, 29, 31, 2, 37, 41, 43, 47, 7, 53, 59, 61, 2, 67, 71, 73, 79, 3, 83, 89, 97, 101, 103, 107, 109, 113, 11, 5, 127, 2, 131, ...}

A063872 Let m be the n-th, n ≥ 1, positive integer such that phi(m) is divisible by m - phi(m). Then a(n) = phi(m)/(m - phi(m)).
(LCM numeral system: A063872(n+1) is maximum digit for index n, n ≥ 0; A063872(n) is maximum digit for index -n, n ≥ 1.)

{1, 2, 1, 4, 6, 1, 2, 10, 12, 1, 16, 18, 22, 4, 2, 28, 30, 1, 36, 40, 42, 46, 6, 52, 58, 60, 1, 66, 70, 72, 78, 2, 82, 88, 96, 100, 102, 106, 108, 112, 10, 4, 126, 1, 130, ...}

A051451 LCM{ 1,2,...,x } where x is a prime power (A000961).
(LCM numeral system: A051451(n+1) is place value for index n, n ≥ 0; A051451(-n+1) is (place value)^(-1) for index n, n < 0.)

{1, 2, 6, 12, 60, 420, 840, 2520, 27720, 360360, 720720, 12252240, 232792560, 5354228880, 26771144400, 80313433200, 2329089562800, 72201776446800, ...}

### Table

"LCM representation" of nonnegative integers from 0 to LCM(1, ..., 5) − 1

n10 nLCM
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 1:0:0:0
13 1:0:0:1
14 1:0:1:0
15 1:0:1:1
16 1:0:2:0
17 1:0:2:1
18 1:1:0:0
19 1:1:0:1
20 1:1:1:0
21 1:1:1:1
22 1:1:2:0
23 1:1:2:1
24 2:0:0:0
25 2:0:0:1
26 2:0:1:0
27 2:0:1:1
28 2:0:2:0
29 2:0:2:1

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

## Representation of rational numbers

This may be the "smallest" product-based numbering system that has a unique finite representation for every rational number. In this base 1/2 = .1 (1*1/2), 1/3 = .02 (0*1/2 + 2*1/6), 1/5 = .0102 (0*1/2 + 1*1/6 + 0*1/12 + 2*1/60).Russell Easterly, Oct 03 2001

The following table shows how to proceed to get a unique representation for all nonnegative rational numbers from the (0, 1) open unit interval. (This is definitively a very unwieldy numeral system: you have to be very careful about the range of digits for negative index ${\displaystyle k}$, which goes from ${\displaystyle 0}$ up to ${\displaystyle {\tfrac {1}{r_{k}}}-1}$, where ${\displaystyle r_{k}}$ is the radix for index ${\displaystyle k}$.)

 index: −1 −2 −3 −4 −5 −6 −7 radix: 1/2 1/3 1/2 1/5 1/7 1/2 1/3 place value: 1/LCM(1, .., 2) = 1/2 1/LCM(1, .., 3) = 1/6 1/LCM(1, .., 2^2) = 1/12 1/LCM(1, .., 5) = 1/60 1/LCM(1, .., 7) = 1/420 1/LCM(1, .., 2^3) = 1/840 1/LCM(1, .., 3^2) = 1/2520 max weight 1 × 1/2 = 1/2 2 × 1/6 = 1/3 1 × 1/12 = 1/12 4 × 1/60 = 1/15 6 × 1/420 = 1/70 1 × 1/840 = 1/840 2 × 1/2520 = 1/1260 max value 1/2 = 1/2 1/2 + 2/6 = 5/6 5/6 + 1/12 = 11/12 11/12 + 4/60 = 59/60 59/60 + 6/420 = 419/420 419/420 + 1/840 = 839/840 839/840 + 2/2520 = 2519/2520 digit: 0 to 1 0 to 2 0 to 1 0 to 4 0 to 6 0 to 1 0 to 2