Sequence A073733, Convergents to log210
In computer technology there is the commonly used "K" approximation 210 = 1024 ≈ 1000 = 103. This conincidence is convenient to users referring to the sizes of files and other computer things like memory and hard drives, and it results from the fact that the base-10 logarithm of 2 is very close to 3/10.
There are higher powers of 2 that make better approximations to a power of 10, but none is particularly convenient. They can be found using a continued fraction series (see A028232) to log210 = 3.32192809488... The numerators of the fractions are sequence A073733, which give the powers of 2 that approximate powers of 10. The first few of these approximations are:
210 = 1024 ≈ 103
293 = 9.9035203142...×1027 ≈ 1028
2196 = 1.0043362776...×1059 ≈ 1059
2485 = 9.9895953610...×10145 ≈ 10146
22136 = 1.000162894...×10643 ≈ 10643
213301 = 9.99936281...×104003 ≈ 104004
...
The powers of 10 are found in sequence A046104.
Some other sequences are discussed here.
This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2013 Mar 11.
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