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:

 1       1476395008 = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*22
 2     116508327936 = 4*4*4*4*4*4*4*4*4*444444
 3     505627938816 = 4*4*4*4*4444*444444
 4     640532803911 = 7*7*7*7*7*7*7*777777
 5    1207460451879 = 3*33*33*333*333*3333
 6    1429150367744 = 8*8*8*8*8*8*8*88*88*88
 7    1458956660623 = 7*77*77*77*77*77*77
 8    3292564845031 = 7*7777*7777*7777
 9    3820372951296 = 44*44*444*4444444
10    5056734498816 = 2*2*2*2*2*2*2*2*2*2*22222*222222
11    6784304541696 = 2*2*2*2*2*2*2*22*22*222*222*2222
12    8090702381056 = 4*4*4*4*4*4*44444*44444
13    9095331446784 = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*22*222*222
14   10757095489536 = 2*2*2*2*2*2*2*2*2*2*2*22*22*22*222*2222
15   10973607685048 = 22222*22222*22222
16   13505488366293 = 7*7*77*77*77*777*777
17   14913065975808 = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*222222
18   38203732951296 = 44*44*444*44444444
19   44859347140608 = 2*2*2*2*2*2*2*2*2*2*2*2*222*222*222222
20   50567390498816 = 2*2*2*2*2*2*2*2*2*2*22222*2222222
21   52612606387341 = 9*9*9*9*9*9*99*999999
22   76259892101481 = 3*3*3*3*3*3*3*3*3*3*33*33*33*33*33*33
23   88990517231616 = 4*4*4*4*4*44*4444*444444
24   89405043019776 = 2*2*2*2*22*22*22*22*22*22*222*222
25   97801459531776 = 2*2*2*2*2*2*2*2*2*2*2*2*22*22*222*222222
26  109737064485048 = 22222*22222*222222
27  119706531338304 = 222*222*222*222*222*222
28  124004938635963 = 7*7*7*77*777*777*7777
29  130043698937856 = 2*2*2*2*2*2*2*2*2*22*22*22*22*22*222*222
30  141759347490816 = 2*2*2*2*2*2*2*2*22*22*22*22*22*22*22*222
31  154530459877376 = 2*2*2*2*2*2*2*22*22*22*22*22*22*22*22*22
32  187619251060736 = 4*4*4*4*44*44*44*44*44*4444
33  191190753643648 = 2*2*2*22*22*2222*22222222

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 √Čric 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.