  For $b>1$ and $\alpha\beta$ a string of two digits in base $b$, let $K_1$ be
the subsum of the harmonic series with only those integers having exactly one
occurrence of $\alpha\beta$. We obtain a theoretical representation of such
$K_1$ series which, say for $b=10$, allows computing them all to thousands of
digits. This is based on certain specific measures on the unit interval and the
use of their Stieltjes transforms at negative integers. Integral identities of
a combinatorial nature both explain the relation to the $K_1$ sums and lead to
recurrence formulas for the measure moments allowing in the end the
straightforward numerical implementation.
