## Monday, April 22, 2024

### Blossoms

It's the height of Toronto's cherry blossoms today, supposedly. It's also the start of our outside maple trees blossoming, which means soon enough they'll be littering the ground and it'll be impossible not to trek them into the house!

## Friday, April 19, 2024

### Ed Pegg's product partition challenge

Now that Ed Pegg's recent Math-Fun suggestion is ensconced in the OEIS, I will highlight his assertion that the smallest product with a single-digit factorization is 1476395008. My idea is to enumerate a bunch of such integers by multiplying together all possible combinations of all possible powers of repdigits (of 2, 3, 4, 7, 8, 9), ignoring numbers larger than some limit. The products are then examined for having the nine digits that are not the factorization digit.

I managed to generate 2554 terms (<10^24) before running out of RAM. Michael Branicky upped this to 10000 terms (available as a b-file in OEIS A372106). Here is how things start:

I had put the fully factored 10000 terms here but Neil saw fit to add it to the OEIS.

## Saturday, April 13, 2024

### Pandigital products

Based on a Neil Sloane misreading of an Ed Pegg idea (link has 187511 products, 10 MB):

8596 = 2*14*307

8790 = 2*3*1465

9360 = 2*4*15*78

9380 = 2*5*14*67

9870 = 2*3*1645

10752 = 3*4*896

12780 = 4*5*639

14760 = 5*9*328

14820 = 5*39*76

15628 = 4*3907

15678 = 39*402

16038 = 27*594 = 54*297

16704 = 9*32*58

17082 = 3*5694

17820 = 36*495 = 45*396

17920 = 8*35*64

18720 = 4*5*936

19084 = 52*367

19240 = 8*37*65

20457 = 3*6819

20574 = 6*9*381

20754 = 3*6918

21658 = 7*3094

24056 = 8*31*97

24507 = 3*8169

25803 = 9*47*61

26180 = 4*7*935

26910 = 78*345

27504 = 3*9168

28156 = 4*7039

28651 = 7*4093

30296 = 7*8*541

30576 = 8*42*91

30752 = 4*8*961

31920 = 5*76*84

32760 = 8*45*91

32890 = 46*715

34902 = 6*5817

36508 = 4*9127

47320 = 8*65*91

58401 = 63*927

65128 = 7*9304

65821 = 7*9403

Neil has fast-tracked this into the OEIS.

## Monday, April 08, 2024

### A Falcoz digit-fancy

click to enlarge |

In addition to my own "fanciful extension" of Éric Angelini's Two identical digits effort, Jean-Marc Falcoz suggested his own variation (at the end of the blog entry): "Lexicographically earliest sequence of distinct positive terms such that [the product of adjacent terms] contains exactly 1 digit 1 (if 1 is present), 2 digits 2 (if 2 is present), 3 digits 3 (if 3 is present), ... 9 digits 9 (if 9 is present)." He presented 113 terms of the sequence but I was hungry for more.

The above plot just exceeds 1000 terms. Surprisingly, term #318 is 17 and term #319 is 13, local minima. Term #455 is 1011211671, a thus-far maximum. Possible products are given by A108571. Our indexed products are such that product #2 is term #2 multiplied by term #1 (product #1 is 1 by fiat). In the current list there is only one duplicate: product #172 = product #622 = 2423433144. If *typed* by their constituent digits, regardless of digit order, the number of possible types is given by A125573. Our current list realizes just 71, the number of which (sorted by product digit-length) are: 1, 1, 1, 2, 0, 0, 2, 3, 2, 4, 5, 7, 9, 11, 9, 10, 4.