## 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.