It has for many years been popular to use RSA keys that have lengths that are powers of two. E.g. 1024-bit, 2048-bit and 4096-bit key lengths are all popular for use with OpenPGP implementations such as GnuPG, and with OpenSSH, etc, and these key lengths are often either defaults in the software generating the keys, or are recommended by organisations whose participants are required to generate keys (e.g. Debian).
I have heard it said that, because cryptanalysts in practice will have devoted more resources towards breaking keys with popular lengths, and because popular lengths are powers of two, choosing a key length that is not a power of two would provide a practical security advantage, even if this means choosing a slightly shorter key length than one would otherwise desire (e.g. 4093-bit key instead of 4096-bit).
Is there any sense in that saying? Put another way, what (if any) are the reasons for which a cryptanalytic attack that would succeed against a 4096-bit key would fail or take significantly longer against a shorter key whose length is not a power of two?