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.
οὔτοι ἀπ’ ἀρχῆς πάντα θεοὶ θνητοῖς ὑπέδειξαν,
ἀλλὰ χρόνῳ ζητέοντες ἐφευρίσκουσιν ἄμεινον. (Xenophanes, Fragment 16)
Carl W. Conrad
Department of Classics, Washington University (Retired)