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<p class=MsoNormal>I was thinking of an improvement to the OEIS logarithmic
graphs.<o:p></o:p></p>
<p class=MsoNormal><o:p> </o:p></p>
<p class=MsoNormal>I acknowledge the importance of the log graphs. For example,
if a sequence has extremely fast growth, the regular plot is unrevealing, as
most of the elements are plotted at 0. An example is A000142, where the normal
plot is pretty much useless, while the log plot nicely reveals smooth mildly
superexponential growth.<o:p></o:p></p>
<p class=MsoNormal><o:p> </o:p></p>
<p class=MsoNormal>While the current log plots are very useful, I do have a
couple of peeves.<o:p></o:p></p>
<p class=MsoNormal><o:p> </o:p></p>
<p class=MsoNormal>My minor peeve is that the log plots are inconsistent.
Either log(|a(n)|+1) or log(|a(n)|) and sometimes just n is plotted depending
on the sequence’s range and whether or not it includes 0. For instance,
compare A000142, A000179 and A008683.<o:p></o:p></p>
<p class=MsoNormal><o:p> </o:p></p>
<p class=MsoNormal>My second peeve is a little more peevish. Specifically, when
a sequence includes both large positive and negative values, these values are
both plotted in the positive direction. This is particularly troublesome when
large positive and negative values are closely interleaved. For instance, the
log plots of A039834 and A053495 are rather deceiving.<o:p></o:p></p>
<p class=MsoNormal><o:p> </o:p></p>
<p class=MsoNormal>I have a suggestion for improving the log plots.<o:p></o:p></p>
<p class=MsoNormal><o:p> </o:p></p>
<p class=MsoNormal>I suggest that instead of log10(x), we do log plots with the
quasi-log10 function<o:p></o:p></p>
<p class=MsoNormal><o:p> </o:p></p>
<p class=MsoNormal> [1] qlog10(x) =
log((x+sqrt(x^2+4))/2)/log(10).<o:p></o:p></p>
<p class=MsoNormal><o:p> </o:p></p>
<p class=MsoNormal>qlog10 satisfies<o:p></o:p></p>
<p class=MsoNormal><o:p> </o:p></p>
<p class=MsoNormal> [2] qlog10(10^x-10^(-x)) = x.<o:p></o:p></p>
<p class=MsoNormal><o:p> </o:p></p>
<p class=MsoNormal>This is, q is the inverse of x -> 10^x-10^(-x). Thus q is
a nice smooth increasing R->R function.<o:p></o:p></p>
<p class=MsoNormal><o:p> </o:p></p>
<p class=MsoNormal>From [2], we obtain some nice graphical properties of
qlog10.<o:p></o:p></p>
<p class=MsoNormal><o:p> </o:p></p>
<p class=MsoNormal> [3] qlog10(-x) = -qlog10(x).<o:p></o:p></p>
<p class=MsoNormal><o:p> </o:p></p>
<p class=MsoNormal>qlog10 is an odd function. positive, negative, and zero a(n)
will map to positive, negative and zero y-coordinates, respectively. [3] also
allows us to implement<o:p></o:p></p>
<p class=MsoNormal><o:p> </o:p></p>
<p class=MsoNormal> [4] qlog10(x) = [1] if x >= 0;
-qlog10(-x) if x < 0<o:p></o:p></p>
<p class=MsoNormal><o:p> </o:p></p>
<p class=MsoNormal>which avoids unstable calculations of [1] on large negative
arguments.<o:p></o:p></p>
<p class=MsoNormal><o:p> </o:p></p>
<p class=MsoNormal>We also have<o:p></o:p></p>
<p class=MsoNormal><o:p> </o:p></p>
<p class=MsoNormal> [5] For x >= 1, 0 <
qlog10(x)-log10(x) < log10(2)/x^2.<o:p></o:p></p>
<p class=MsoNormal><o:p> </o:p></p>
<p class=MsoNormal>In other words, qlog10(x) is close enough to log10(x) for
graphing purposes. Specifically, qlog10(10) = 1.00428..., a perturbation of
less that 1/2 %, which means that 10^n will visually map very closely to y-axis
tick mark n for n >= 1. Even qlog10(1) = .20898+ is not too far from 0. qlog10(x)
is definitely more accurate than log(x)+1, currently used for many sequences.<o:p></o:p></p>
<p class=MsoNormal><o:p> </o:p></p>
<p class=MsoNormal>To summarize, plotting qlog10(n) gives a plot that preserves
order and sign and closely approximates sgn(n)*log10(n) fairly well for all but
the smallest n, and which can be used to plot any sequence regardless of its
range.<o:p></o:p></p>
<p class=MsoNormal><o:p> </o:p></p>
<p class=MsoNormal>For someone with plotting capabilities, it might be
instructive to create a graph which compares log10(x), -log10(-x) and
qlog10(x).<o:p></o:p></p>
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