Welcome to the Playful Math Carnival, 155th edition!
155, tell us your secrets.
Via Pat Bellew, 155 is the sum of the prime numbers between its smallest and largest prime factors, 5 and 31. 5+7+11+13+17+19+23+29+31=155. How would you go about finding more of these? What would you call them? Pat also notes that 155 is the number of primitive permutation groups of order 81. Which is odd, because it is more than double the number of groups for any order less than 81. And there's not another larger (than 75 even!) until you get to order 256 (which has 244). Do 81 and 256 have anything in common?
Wait, 5 and 31? That means 155 is semiprime. What is the previous and what is the next semiprime? (They're both even...) Are there more primes or semiprimes smaller than 100?
The coolest thing I found is that 155 is a toothpick number. You start with a toothpick, then add a perpendicular toothpick anywhere there is an exposed endpoint. Here is 1, 3, 7, 11, 15, 23, 35, 43, 47, 55, 67. How many more steps to 155? Is it a fractal? Is it a cellular automaton? Mathematicians have also studied T(n)/n^2. Does it have a limit? Does it have an extremum? Here's some GeoGebra to make your own.
155 is also a generalized pentagonal number. The pentagonal numbers have a rule n(3n-1)/2, usually for n =1, 2, 3... , giving 1, 5, 12, 22, 35, ... But there are also positive outputs for negative integers, 2, 7, 15, 26, 40 ... which pleasantly fit between the usual pentagonal numbers. What patterns do you notice? Which negative number gives 155? I've been trying to think about how to visualize these negative pentagonals, to no avail so far. Have you got any ideas?
Maybe the toothpick was a little too crazy of a visual patten? Here's one I was trying to make to have 155. Did it work? If so, which step? Fawn always asks for the 43rd step... what's that? Is there a rule? What if step 1 had -1 square, what would the rule be?
The previous Playful Math Carnival was at Denise Gaskins' blog, the founder of the carnival. Be sure to check her site weekly for the Math Game Mondays which are only up one week! Other goodies, too, though. Next up is at Nature Study Australia. Contact Denise if you're interested or willing to host. It really impresses me every time I do just how much good stuff is out there.
PS. I've been working all year with Xavier Golden (yes relation) a preservice art teacher on a math graphic novel. And we're starting to see some inked and colored pages... I'm so excited!