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LCM numeral system
The (tentatively named) LCM numeral system^{[1]} uses the least common multiple (LCM) of the first prime powers as place value for index , where the first prime power is the place value for index 0.
Contents
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 , which goes from up to , where is the radix for index .)
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 nth 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 nth, 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


Representation of rational numbers
This may be the "smallest" productbased 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 , which goes from up to , where is the radix for index .)
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 
See also
Notes
 ↑ The (tentatively named) LCM numeral system would be more precisely named prime powers LCM numeral system, although a bit long... did anyone find/propose any name for this numeral system?