# Six Six Nix

4 x 3 = 13. A mistake? Not in base-9, where 13 = 1×9^1 + 3 = 12 in base-10. This means that 13 is a sum-product number in base-9: first add its digits, then multiply them, then multiply the digit-sum by the digit-product: (1+3) x (1×3) = 13. There are four more sum-product numbers in this base:

2086 = 17 x 116 = (2 + 8 + 6) x (2 x 8 x 6) = 1536 = 16 x 96
281876 = 35 x 7333 = (2 + 8 + 1 + 8 + 7 + 6) x (2 x 8 x 1 x 8 x 7 x 6) = 172032 = 32 x 5376
724856 = 35 x 20383 = (7 + 2 + 4 + 8 + 5 + 6) x (7 x 2 x 4 x 8 x 5 x 6) = 430080 = 32 x 13440
7487248 = 44 x 162582 = (7 + 4 + 8 + 7 + 2 + 4 + 8) x (7 x 4 x 8 x 7 x 2 x 4 x 8) = 4014080 = 40 x 100352

And that’s the lot, apart from the trivial 0 = (0) x (0) and 1 = (1) x (1), which are true in all bases.

135 = 9 x 15 = (1 + 3 + 5) x (1 x 3 x 5)
144 = 9 x 16 = (1 + 4 + 4) x (1 x 4 x 4)
1088 = 17 x 64 = (1 + 8 + 8) x (1 x 8 x 8)

1088 is missing from the list at Wikipedia and the Encyclopedia of Integer Sequences, but I like the look of it, so I’m including it here. Base-11 has five sum-product numbers:

419 = 13 x 33 = (4 + 1 + 9) x (4 x 1 x 9) = 504 = 14 x 36
253 =  x 28 = (2 + 5 + 3) x (2 x 5 x 3) = 300 = 10 x 30
2189 = 19 x 121 = (2 + 1 + 8 + 9) x (2 x 1 x 8 x 9) = 2880 = 20 x 144
7634 = 19 x 419 = (7 + 6 + 3 + 4) x (7 x 6 x 3 x 4) = 10080 = 20 x 504
82974 = 28 x 3036 = (8 + 2 + 9 + 7 + 4) x (8 x 2 x 9 x 7 x 4) = 120960 = 30 x 4032

But the record for bases below 50 is set by 7:

22 = 4 x 4 = (2 + 2) x (2 x 2) = 16 = 4 x 4
505 = 13 x 34 = (5 + 5) x (5 x 5) = 250 = 10 x 25
242 = 11 x 22 = (2 + 4 + 2) x (2 x 4 x 2) = 128 = 8 x 16
1254 = 15 x 55 = (1 + 2 + 5 + 4) x (1 x 2 x 5 x 4) = 480 = 12 x 40
2343 = 15 x 132 = (2 + 3 + 4 + 3) x (2 x 3 x 4 x 3) = 864 = 12 x 72
116655 = 33 x 2424 = (1 + 1 + 6 + 6 + 5 + 5) x (1 x 1 x 6 x 6 x 5 x 5) = 21600 = 24 x 900
346236 = 33 x 10362 = (3 + 4 + 6 + 2 + 3 + 6) x (3 x 4 x 6 x 2 x 3 x 6) = 62208 = 24 x 2592
424644 = 33 x 11646 = (4 + 2 + 4 + 6 + 4 + 4) x (4 x 2 x 4 x 6 x 4 x 4) = 73728 = 24 x 3072

And base-6? Six Nix. There are no sum-product numbers unique to that base (to the best of my far-from-infallible knowledge). Here is the full list for base-3 to base-50 (not counting 0 and 1 as sum-product numbers):

 5 in base-11 4 in base-21 3 in base-31 2 in base-41 4 in base-12 5 in base-22 1 in base-32 3 in base-42 0 in base-3 3 in base-13 4 in base-23 3 in base-33 4 in base-43 2 in base-4 3 in base-14 2 in base-24 4 in base-34 5 in base-44 1 in base-5 2 in base-15 3 in base-25 2 in base-35 6 in base-45 0 in base-6 2 in base-16 6 in base-26 2 in base-36 7 in base-46 8 in base-7 6 in base-17 0 in base-27 3 in base-37 3 in base-47 1 in base-8 5 in base-18 1 in base-28 3 in base-38 7 in base-48 5 in base-9 7 in base-19 0 in base-29 1 in base-39 5 in base-49 3 in base-10 3 in base-20 2 in base-30 2 in base-40 3 in base-50

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