Perhaps this has already been answered to the satisfaction of Mike. I don't really understand what the problem is. The verb γράφειν has a 1st perfect γέγραφα and a 2nd perfect γεγράφηκα, just as it has a 1st passive ἐγράφθην and a 2nd passive ἐγράφην. The perfect infinitive ending is -έναι. I would expect that one might find both forms in this era when some are beginning to Atticize -- and Josephus falls into that category, doesn't he? That being the case, he would choose the older perfect stem.
It's always seemed a bit amusing to me that the older aorists, perfect, and passives are called "second" and the newer forms are called "first." I realize that "first" here must mean "more common" -- but it still seems to me that the earlier form ought to be called "first" and the later one "second." I remember having to do a second take with Aristotle's distinction of two senses of πρῶτος: "first" in the natural order, and "first" in relation to ourselves.
Carl W. Conrad
Department of Classics, Washington University (Retired)
… ἐπειδὴ καὶ τὸν οἶνον ἠξίους
πίνειν, συνεκποτέ’ ἐστί σοι καὶ τὴν τρύγα Aristophanes, Plutus 1085