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A292275
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A sequence of rounded numbers useful for entering values over several orders of magnitude in computer-human interfaces, with 10 values per order of magnitude.
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0
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100, 125, 150, 200, 250, 300, 400, 500, 600, 800, 1000, 1250, 1500, 2000, 2500, 3000, 4000, 5000, 6000, 8000, 10000, 12500, 15000, 20000, 25000, 30000, 40000, 50000, 60000, 80000, 100000, 125000, 150000, 200000, 250000, 300000, 400000, 500000, 600000, 800000, 1000000
(list;
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refs;
listen;
history;
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internal format)
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OFFSET
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20,1
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COMMENTS
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Values from the real-valued sequence R = {1.0, 1.25, 1.5, 2.0, 2.5, 3.0, 4.0, 5.0, 6.0, 8.0, 10.0, 12.5, 15.0, 20.0, 25.0, 30.0, 40.0, 50.0, 60.0, 80.0, 100.0, 125.0, 150.0, 200.0, 250.0, 300.0, 400.0, 500.0, 600.0, 800.0, 1000.0, ...} are used in certain computer applications, such as geographic information system (GIS) applications where they are provided as round numbers for selection as map scale values. This real-valued sequence R (all of whose values above 12.5 are integers) represents a convenient balance between roundness of the base-10 values and evenness of their spacing (in logarithmic terms).
The real-valued sequence can be continued infinitely in both directions; for simplicity, the terms listed in the Data section for this integer sequence begin at a(20) = 100 = 10^2. (Extending the sequence to lower values of n would cause the noninteger value 12.5 to be reached at n=11.)
Some properties of the sequence (see Example section):
(1) on a logarithmic scale, the terms are fairly evenly spaced;
(2) all terms are round numbers; other than those terms that begin with digits 125, 15, or 25 (each of which has no prime factor larger than 5), each term has only one nonzero digit;
(3) there are 10 terms per order of magnitude;
(4) every ratio between consecutive terms is one of three small fractions: 4/3, 5/4, and 6/5.
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LINKS
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FORMULA
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a(n) = 10^n * 1 if n mod 10 = 0;
10^n * 5/4 if n mod 10 = 1;
10^n * 3/2 if n mod 10 = 2;
10^n * 2 if n mod 10 = 3;
10^n * 5/2 if n mod 10 = 4;
10^n * 3 if n mod 10 = 5;
10^n * 4 if n mod 10 = 6;
10^n * 5 if n mod 10 = 7;
10^n * 6 if n mod 10 = 8;
10^n * 8 if n mod 10 = 9.
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EXAMPLE
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n a(n) a(n)/a(n-1) log_10(a(n)) log_10(a(n)) - n/10
== ==== =========== =============== ===================
20 100 5/4 2.0000000000... 0.0000000000000...
21 125 5/4 2.0969100130... -0.0030899869919...
22 150 6/5 2.1760912590... -0.0239087409443...
23 200 4/3 2.3010299956... +0.0010299956639...
24 250 5/4 2.3979400086... -0.0020599913279...
25 300 6/5 2.4771212547... -0.0228787452803...
26 400 4/3 2.6020599913... +0.0020599913279...
27 500 5/4 2.6989700043... -0.0010299956639...
28 600 6/5 2.7781512503... -0.0218487496163...
29 800 4/3 2.9030899869... +0.0030899869919...
30 1000 5/4 3.0000000000... 0.0000000000000...
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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