A 2-adic formula for Bernoulli numbers of the second kind and for the Nörlund numbers

We give a formula expressing Bernoulli numbers of the second kind as 2-adically convergent sums of traces of algebraic integers. We use this formula to prove and explain the formulas and conjectures of Adelberg concerning the initial 2-adic digits of these numbers. We also give analogous results for the Nörlund numbers.