Saturday, May 02, 2020

Only eight

This typewritten table of the decimal expansions of powers of two up to 2^115 dates to when I was fourteen years old. My fascination then with powers of two almost certainly arose as a consequence of having encountered the wheat and chessboard problem. I would have calculated the numbers by hand and the typing layout suggests a slight obsession with presentation decorum, a handicap I've endured to the present day. The digit after the power is the digital root.

I recently had occasion to extend OEIS sequence A305942, the number of decimal powers of two having exactly n digits zero. For any given n, that number is fairly constant (on average a little over 33) but there is significant variation. For n up to 295000, I have found a zero-count as high as 62 and as low as 11. Checking other digits in the same range, I find a high of 65 and a low of 8 (see below). These extrema are outliers of course and statistics might suggest that we can find larger-than-65 and smaller-than-8 examples, if only we chart n large enough. But bear in mind that my current database of n up to 295000 is based on powers-of-two decimal expansions up to 2^10000000. It is not a fast computation.

This graph (click on it to get a better view) shows the number of occurrences (the blue points) of the digit 7 in decimal powers of two from 9100000 to 9240000. The green line represents the value 275923. Although (due to the size of the points and the thickness of the line) it may seem that there are dozens of points on the line, there are in fact only eight (at powers 9141747, 9143624, 9155434, 9163531, 9168298, 9171371, 9174454, and 9190491).