The odd part of 2^e! as e → ∞ leads to a 2-adic integer z. The bits of z were publicized in OEIS sequence A359349, where two conjectures were made, relevant to computing z. We prove both of those conjectures. A second 2-adic integer, the limit of ((2^e-1)!!–1)/2^e, plays a key role in one proof.