%I #6 Sep 12 2017 21:34:05
%S 100,125,150,200,250,300,400,500,600,800,1000,1250,1500,2000,2500,
%T 3000,4000,5000,6000,8000,10000,12500,15000,20000,25000,30000,40000,
%U 50000,60000,80000,100000,125000,150000,200000,250000,300000,400000,500000,600000,800000,1000000
%N 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.
%C 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).
%C 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.)
%C Some properties of the sequence (see Example section):
%C (1) on a logarithmic scale, the terms are fairly evenly spaced;
%C (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;
%C (3) there are 10 terms per order of magnitude;
%C (4) every ratio between consecutive terms is one of three small fractions: 4/3, 5/4, and 6/5.
%F a(n) = 10^n * 1 if n mod 10 = 0;
%F 10^n * 5/4 if n mod 10 = 1;
%F 10^n * 3/2 if n mod 10 = 2;
%F 10^n * 2 if n mod 10 = 3;
%F 10^n * 5/2 if n mod 10 = 4;
%F 10^n * 3 if n mod 10 = 5;
%F 10^n * 4 if n mod 10 = 6;
%F 10^n * 5 if n mod 10 = 7;
%F 10^n * 6 if n mod 10 = 8;
%F 10^n * 8 if n mod 10 = 9.
%e n a(n) a(n)/a(n-1) log_10(a(n)) log_10(a(n)) - n/10
%e == ==== =========== =============== ===================
%e 20 100 5/4 2.0000000000... 0.0000000000000...
%e 21 125 5/4 2.0969100130... -0.0030899869919...
%e 22 150 6/5 2.1760912590... -0.0239087409443...
%e 23 200 4/3 2.3010299956... +0.0010299956639...
%e 24 250 5/4 2.3979400086... -0.0020599913279...
%e 25 300 6/5 2.4771212547... -0.0228787452803...
%e 26 400 4/3 2.6020599913... +0.0020599913279...
%e 27 500 5/4 2.6989700043... -0.0010299956639...
%e 28 600 6/5 2.7781512503... -0.0218487496163...
%e 29 800 4/3 2.9030899869... +0.0030899869919...
%e 30 1000 5/4 3.0000000000... 0.0000000000000...
%Y Cf. A231848.
%K nonn
%O 20,1
%A _Jon E. Schoenfield_, Sep 12 2017