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!

maple blossoms: April 25

maple blossoms give way to leaves: May 4

fallen maple blossoms: May 4

more fallen maple blossoms: May 6

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.

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.

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.

My initial plot (above) just exceeds 1000 terms. My updated (May 13) plot (below) extends this to 5000 terms. Term #4367 = 1785221551, a local maximum. Term #2213 = 2. Also known to appear are 8, 11, 13, 17, 22, 44, 84, 97, ... Possible products are 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 product list we have 16 duplicates and 2 triplicates (#3 = #2215 = #2219 and #416 = #2349 = #3632). If typed by their constituent digits, regardless of digit order, the number of possible types is given by A125573. Our current list realizes 105 of these, the number of which (sorted by product digit-length) are: 1, 1, 1, 2, 1, 3, 4, 4, 6, 7, 8, 11, 10, 14, 13, 11, 7, 1.

click to enlarge