The buchstab-function
https://en.wikipedia.org/wiki/Buchstab_function
is a delayed differential equation defined as follows :
For $1\le u\le 2$ we have $\omega(u)=\frac{1}{u}$
For $u\ge 2$ we have $(u\omega(u))'=\omega(u-1)$
How can I calculate the buchstab-function for some value $u$ numerically ?
A table with some special values would also be fine.
I would like to have an algorithm working in PARI/GP or an online-calculator doing that :
I have neither Mathematica nor Matlab.
A plot of the Buchstab function is:
Below is a table of values in $0.05$ increments for $u \in [1, 6]$. Notice in the plot and data that the Buchstab function approaches ${\displaystyle e^{-\gamma }} $ rapidly as ${\displaystyle u\to \infty ,} $ , where ${\displaystyle \gamma }$ is the Euler–Mascheroni constant. The numerical value is $0.5614594835668852$.
You can download a free Computer Algebra System or Octave (a free variant of Matlab) that supports routines for DDEQ and code it up. Python also has routines to support such things.
$$\begin{array}{|c|c|} \text{u} & \text{w(u)} \\ \hline 1.&1.\\ \hline 1.05&0.952381\\ \hline 1.1&0.909091\\ \hline 1.15&0.869565\\ \hline 1.2&0.833333\\ \hline 1.25&0.8\\ \hline 1.3&0.769231\\ \hline 1.35&0.740741\\ \hline 1.4&0.714286\\ \hline 1.45&0.689655\\ \hline 1.5&0.666667\\ \hline 1.55&0.645161\\ \hline 1.6&0.625\\ \hline 1.65&0.606061\\ \hline 1.7&0.588235\\ \hline 1.75&0.571429\\ \hline 1.8&0.555556\\ \hline 1.85&0.540541\\ \hline 1.9&0.526316\\ \hline 1.95&0.512821\\ \hline 2.&0.5\\ \hline 2.05&0.511605\\ \hline 2.1&0.521576\\ \hline 2.15&0.530122\\ \hline 2.2&0.537419\\ \hline 2.25&0.543619\\ \hline 2.3&0.548854\\ \hline 2.35&0.553236\\ \hline 2.4&0.556863\\ \hline 2.45&0.559822\\ \hline 2.5&0.562186\\ \hline 2.55&0.564022\\ \hline 2.6&0.565386\\ \hline 2.65&0.56633\\ \hline 2.7&0.566899\\ \hline 2.75&0.567133\\ \hline 2.8&0.567067\\ \hline 2.85&0.566732\\ \hline 2.9&0.566157\\ \hline 2.95&0.565366\\ \hline 3.&0.564382\\ \hline 3.05&0.563424\\ \hline 3.1&0.562671\\ \hline 3.15&0.562088\\ \hline 3.2&0.561647\\ \hline 3.25&0.561324\\ \hline 3.3&0.561096\\ \hline 3.35&0.560947\\ \hline 3.4&0.560861\\ \hline 3.45&0.560826\\ \hline 3.5&0.560829\\ \hline 3.55&0.560861\\ \hline 3.6&0.560915\\ \hline 3.65&0.560984\\ \hline 3.7&0.56106\\ \hline 3.75&0.56114\\ \hline 3.8&0.561219\\ \hline 3.85&0.561293\\ \hline 3.9&0.561359\\ \hline 3.95&0.561415\\ \hline 4.&0.561458\\ \hline 4.05&0.561488\\ \hline 4.1&0.561507\\ \hline 4.15&0.561517\\ \hline 4.2&0.561521\\ \hline 4.25&0.561521\\ \hline 4.3&0.561517\\ \hline 4.35&0.561511\\ \hline 4.4&0.561504\\ \hline 4.45&0.561497\\ \hline 4.5&0.56149\\ \hline 4.55&0.561482\\ \hline 4.6&0.561476\\ \hline 4.65&0.56147\\ \hline 4.7&0.561466\\ \hline 4.75&0.561462\\ \hline 4.8&0.561459\\ \hline 4.85&0.561457\\ \hline 4.9&0.561455\\ \hline 4.95&0.561455\\ \hline 5.&0.561454\\ \hline 5.05&0.561455\\ \hline 5.1&0.561455\\ \hline 5.15&0.561456\\ \hline 5.2&0.561456\\ \hline 5.25&0.561457\\ \hline 5.3&0.561457\\ \hline 5.35&0.561458\\ \hline 5.4&0.561458\\ \hline 5.45&0.561459\\ \hline 5.5&0.561459\\ \hline 5.55&0.561459\\ \hline 5.6&0.56146\\ \hline 5.65&0.56146\\ \hline 5.7&0.56146\\ \hline 5.75&0.56146\\ \hline 5.8&0.56146\\ \hline 5.85&0.56146\\ \hline 5.9&0.56146\\ \hline 5.95&0.56146\\ \hline 6.&0.56146\\ \hline \end{array}$$