For each positive integer $p$, there exists a holomorphic curve of order $p$ mean type with an infinite number of deficiencies, the sum of which to the a power is divergent, where $0 < a < 1/3$.
Nobushige Toda. "Holomorphic curves with an infinite number of deficiencies." Proc. Japan Acad. Ser. A Math. Sci. 80 (6) 90 - 95, June 2004. https://doi.org/10.3792/pjaa.80.90