I was surprised to see that the integer sequence 1,3,11,13,17,23,73,... consisting of all the record-breakers in sequence A005589 (i.e., all positive integers whose representation as an English word or phrase is longer than the English representations of all earlier positive integers) isn’t a sequence of its own. Did I make a mistake in computing the list of record-breakers? I actually don’t find such sequences very interesting, but I know others do. In particular, my daughter has asked me to write a blog essay about the fact that repeatedly applying the map k->A005589 <http://oeis.org/A005589>(k) to any starting value n always leads to 4 (cf. A016037 <http://oeis.org/A016037>, A133418 <http://oeis.org/A133418>). Has anyone (perhaps Diane Karloff, who is credited with this observation) written about this? Then maybe I could give the article to my daughter instead of having to write it myself. :-) My daughter and I learned about this fact from my son, who came home from camp last summer challenging us to make sense of baffling conversations like this: Son: Pick another number. Me: Okay, seventy-seven. Son: ... Seventy-seven is twelve, twelve is six, six is three, three is five, five is four, four is the magic number. Have any of you heard this “game” before? Jim Propp