lucy24 - 6:27 pm on May 30, 2013 (gmt 0)
And that's just off the top of my head.
:: shuffling papers ::
126.96.36.199/19 but it tends to stick with a pretty exact IP for quite a while.
Editorial comment: If your hosts don't similarly know the googlebot's IP off the top of their heads, they have possibly not been working hard enough. And the UA string always contains the word "Googlebot", possibly in conjunction with something else, like "-Mobile" or "-Image".
Edit: Now, wait a minute. If the word "Googlebot" doesn't show up in logs at all, then it can only be a DNS issue. But google must be reaching your site, or it wouldn't be indexed. And if weren't indexed, it would not come up in searches.