On 8/30/14, Warren D Smith <warren.wds@gmail.com> wrote:
How many ways PB(N), are there to partition N into powers of 2?
PB(3)=2 PB(5)=4 PB(9)=10 PB(17)=36 PB(33)=202 PB(65)=1828 PB(129)=27338 PB(257)=692004 PB(513)=30251722 PB(1025)=2320518948 PB(2049)=316359580362 PB(4097)=77477180493604 PB(8193)=34394869942983370 PB(16385)=2.78939e+19=N^(0.329570*lgN) PB(32769)=4.16037e+22=N^(0.333950*lgN) PB(65537)=1.14788e+26=N^(0.338160*lgN) PB(131073)=5.8888e+29=N^(0.342193*lgN) PB(262145)=5.64265e+33=N^(0.346049*lgN) PB(524289)=1.01392e+38=N^(0.349732*lgN) PB(1048577)=3.42887e+42=N^(0.353247*lgN) PB(2097153)=2.18936e+47=N^(0.356601*lgN) PB(4194305)=2.64707e+52=N^(0.359803*lgN) PB(8388609)=6.07631e+57=N^(0.362860*lgN) PB(16777217)=2.65451e+63=N^(0.365781*lgN) PB(33554433)=2.21183e+69=N^(0.368573*lgN) PB(67108865)=3.52225e+75=N^(0.371244*lgN) PB(134217729)=1.07397e+82=N^(0.373801*lgN) PB(268435457)=6.28084e+88=N^(0.376251*lgN) PB(536870913)=7.05642e+95=N^(0.378599*lgN) PB(1073741825)=1.52522e+103=N^(0.380853*lgN) PB(2147483649)=6.3513e+110=N^(0.383017*lgN) PB(4294967297)=5.10186e+118=N^(0.385096*lgN) PB(8589934593)=7.91504e+126=N^(0.387096*lgN) ...aha, I see David Wilson found the asymptotic formula in OEIS via N.G. de Bruijn: On Mahler's partition problem, Indagationes Mathematicae, vol. X (1948) 210-220. OK, I'll stop computing now :)