All pertinent results on the divisibility of Bernoulli numbers I have come across in the literature are negative in nature, including what is possibly the only one pertaining to an index of a specific form: that $B_{(p+1)/2}$ is *not* divisible by $p$ if $p \equiv 3 \bmod{4}$. The idea behind this statement traces back to Augustin Cauchy, “Mémoires sur la théorie des nombres,” *Mémoires de l’Académie Royale des Sciences de l’Institut de France* 17 (1840): 249–768, at p. 445, and it is clearly implied in A. Friedmann and J. Tamarkine, “Quelques formules concernant la théorie de la fonction [x] et des nombres de Bernoulli,” *Journal für die reine und angewandte Mathematik* 135 (1909): 146–56, at p. 156 (which however contains a misprint). More modern-style proofs appear in Leonard Carlitz, “The first factor of the class number of a cyclic field,” *Canadian Journal of Mathematics* 6 (1954): 23–26, at p. 25, and in Wells Johnson, “$p$-adic Proofs of Congruences for the Bernoulli Numbers,” *Journal of Number Theory* 7 (1975): 251–265, at p. 257, where it is described as “one of the few general results which seems to be known along these lines.” Stephen V. Ullom, “Upper bounds for $p$-divisibility of sets of Bernoulli numbers,” *Journal of Number Theory* 12 (1980): 197–200, showed that at most one-half of the $B_{2k}$ with $2k \le p-3$ are divisible by $p$, and Ullom’s result is sharpened in Samuel S. Wagstaff, Jr., “$p$-Divisibility of Certain Sets of Bernoulli Numbers,” *Mathematics of Computation* 34 (1980): 647–649.