# Sequence A073733, Convergents to log_{2}10

In computer technology there is the commonly used "K" approximation
2^{10} = 1024 ≈ 1000 = 10^{3}. 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 log_{2}10 =
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:

2^{10} = 1024 ≈ 10^{3}

2^{93} = 9.9035203142...×10^{27} ≈ 10^{28}

2^{196} = 1.0043362776...×10^{59} ≈ 10^{59}

2^{485} = 9.9895953610...×10^{145} ≈ 10^{146}

2^{2136} = 1.000162894...×10^{643} ≈ 10^{643}

2^{13301} = 9.99936281...×10^{4003} ≈ 10^{4004}

...

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 2018 Aug 27. s.11