Sunday, August 28

A factorization balancing act

A couple of weeks ago, Claudio Meller presented 26487 and 65821 as examples of the property of having one each of the base-ten digits when combined with the digits of their respective factorizations. Surprisingly, he missed two:

    26487 = 3^5 * 109
    28651 = 7 * 4093
    61054 = 2 * 7^3 * 89
    65821 = 7 * 9403

I wondered how this might be turned into a sequence. Base-ten k-balanced factorization integers: The combined digits of an integer and its factorization primes and exponents contain exactly k copies of each of the ten digits. So,

 45849660 = 2^2 * 3 * 5 * 19 * 37 * 1087
 84568740 = 2^2 * 3 * 5 * 67 * 109 * 193
104086845 = 3^2 * 5 * 19 * 23 * 67 * 79
106978404 = 2^2 * 3 * 13 * 685759

and so on. For any given k, k-balanced integers are necessarily finite. For k=2, there are more than 13000. Could the largest of these be larger than the smallest 3-balanced integer?

It's not too difficult to generate very large terms:


is an example of a 13-balanced integer. Can you come up with a larger one?

Saturday, August 6

My 100th Leyland prime find

Last October I found my first previously unknown Leyland (probable) prime. Today I found my 100th. The graph shows (in order of discovery) the number of decimal digits of those 100 primes, ranging from 43633 to 61184. The finds have had the unintended (but certainly not unpleasant) consequence of pushing me up a list of probable prime contributors! My search for Leyland primes is of course in aid of my Leyland prime indexing effort which has reached #1137. There'll be a significant jump in about three weeks when I finish rounding up the few remaining unknowns with decimal digit lengths of between 54334 and 55390.

Saturday, July 30

Point of entry

We have a totally fenced yard at the back and sides of the house. To be more accurate, a monster shed on the adjacent property at the back of the yard takes the place of a fence. The front fencing on the (gravel) driveway side accommodates a gate that can be swivelled to allow vehicles into the yard. It is this gate that is being compromised.

For a number of years now some creature has been creating a summer underpass here to allow entry into (and presumably exit from) the yard. I have a shovel nearby to refill the hole and have come to place some small concrete chunks on the yard side of the gate to give the visitor a nasty surprise half-way through its dig. Sometimes a new underpass is generated to one side of the concrete but more generally the creature gives up.

I had always thought the offender was a skunk but some weeks ago Bodie and I were surprised — in broad daylight — by a rat running through the yard, only to disappear (after it was surprised by us) under that back shed!

Friday, July 29


Not too far from our home, this car-crash situation stopped Weston Road traffic both ways for hours. Not seen in the photo are two additional cars to the left, one of which seems to have sustained significant damage. It is difficult for me to imagine a scenario that would flip a car just so but there it is. Notice a woman's shoe sitting on the undercarriage. I haven't found a hint of this event in the media!

Thursday, July 28

Farmers' market

The long-running Weston farmers' market has moved a little closer to my home this year because of impending construction at the old (parking lot) site. I rarely go but decided to accompany Catherine (green hat, far left, below) back on the 16th. It's much smaller than the old site and — in terms of local produce or decent deals — a little disappointing.

Monday, June 20

Cubic Lock

These are the four pieces of Cubic Lock by Goh Pit Khiam as realized in exotic woods by Brian Menold. A union of the two pieces on the left (top, the key made up of 10 cubies; bottom, 9 cubies) is inserted into a union of the two pieces on the right (top, 23 cubies; bottom, 19 cubies) and by means of a strategic to-and-fro of the key, the pieces are shifted into place. It's not obvious that the finished 4*4*4 cube has an internal void until one counts and adds those cubies. A good show-off puzzle as one is unlikely to forget the assembly once one has put it together a few times.

Wednesday, June 8

The wall

I went to Raymore Park (on the other side of the Humber river) last Friday to see what progress had been made on the erosion-control retaining-wall they were putting in place (on my side of the river). Contrary to the intelligence in my previous entry on this, I can now see that the wall will not be so much "on top of the now-in-the-river foundation" as I had supposed but (rather) much higher and more-closely hugging the slope bedrock — which is actually being exposed for a more stable conglomeration. And that storm drain interruption will end up being a barrier to my ever walking along the full length of the wall. At any rate (depending on the wall-top width), it may be too high to be safe. The bits of white floating through the air (more evident in the second photo) is tree fluff.

Monday, May 23

577, 5569, ...

This incipient-sequence comment in T.D. Noe's A138290 recently caught my eye. So, prime p such that 2^(p+1)-2^q-1 is composite for all positive q < p. Carlos Rivera had it published as Puzzle 437 in 2008. Therein, Giovanni Resta notes that he had checked up to 15373 without finding a third term. All of my processors were busy of course but I hadn't been using Catherine's computer in a while (too slow for my needs), so...

This morning (after several weeks of number crunching) I noticed that her machine had come up with the third term: 29251. To give you an idea of the computing involved: on my latest iMac, p = 577 (174-digit numbers) verified compositeness for all terms immediately; p = 5569 (1677-digit numbers) took a minute; and p = 29251 (8806-digit numbers) needed more than four hours! Alas, I'm still one term shy of a new OEIS sequence contribution.

Monday, May 9

Mount Dennis

The part of Toronto in which I live used to be a town called Weston. I live in the south part of Weston, just northwest of a community called Mount Dennis. This photograph was taken in the very northern part thereof, looking down Weston Road towards Toronto's CN Tower. My camera is at maximum zoom, giving the intersection at Jane Street (200 meters away) and the streetscape beyond a seriously foreshortened look. The Tower is 10.8 km distant. I was on my way to Jane's Walk.

Thursday, April 21

Humber river retaining wall

When I speculated one month ago that the current armourstone toe of the valley slope on the east side of the Humber river (not far from my home) was going to be replaced with a higher version, I wasn't entirely correct. It may end up being higher but (more importantly) it will also be further into the river.

Roy Murray continues his Raymore Park updates on the construction (March 30, April 12, April 18, and a follow-up). In the last of these he lets on that the retaining wall will end up on top of the now-in-the-river foundation! This is actually good news for my side of the river because it means that I might be able to access this new stretch of real estate from the south (behind Denison Park) and walk it up to the footbridge (across the river) that lies beyond. Of course Roy worries that the now-narrower river channel will negatively impact his side of the river during flood events. Here's a bird's-eye Apple Maps view of the area (I've put a red square around my house):