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.


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