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_[9]. There are four more sum-product numbers in this base:

2086_[9] = 17 x 116 = (2 + 8 + 6) x (2 x 8 x 6) = 1536_[10] = 16 x 96
281876_[9] = 35 x 7333 = (2 + 8 + 1 + 8 + 7 + 6) x (2 x 8 x 1 x 8 x 7 x 6) = 172032_[10] = 32 x 5376
724856_[9] = 35 x 20383 = (7 + 2 + 4 + 8 + 5 + 6) x (7 x 2 x 4 x 8 x 5 x 6) = 430080_[10] = 32 x 13440
7487248_[9] = 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_[10] = 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.

What about base-10?

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_[11] = 13 x 33 = (4 + 1 + 9) x (4 x 1 x 9) = 504_[10] = 14 x 36
253_[11] = [10] x 28 = (2 + 5 + 3) x (2 x 5 x 3) = 300_[10] = 10 x 30
2189_[11] = 19 x 121 = (2 + 1 + 8 + 9) x (2 x 1 x 8 x 9) = 2880_[10] = 20 x 144
7634_[11] = 19 x 419 = (7 + 6 + 3 + 4) x (7 x 6 x 3 x 4) = 10080_[10] = 20 x 504
82974_[11] = 28 x 3036 = (8 + 2 + 9 + 7 + 4) x (8 x 2 x 9 x 7 x 4) = 120960_[10] = 30 x 4032

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

22_[7] = 4 x 4 = (2 + 2) x (2 x 2) = 16_[10] = 4 x 4
505_[7] = 13 x 34 = (5 + 5) x (5 x 5) = 250_[10] = 10 x 25
242_[7] = 11 x 22 = (2 + 4 + 2) x (2 x 4 x 2) = 128_[10] = 8 x 16
1254_[7] = 15 x 55 = (1 + 2 + 5 + 4) x (1 x 2 x 5 x 4) = 480_[10] = 12 x 40
2343_[7] = 15 x 132 = (2 + 3 + 4 + 3) x (2 x 3 x 4 x 3) = 864_[10] = 12 x 72
116655_[7] = 33 x 2424 = (1 + 1 + 6 + 6 + 5 + 5) x (1 x 1 x 6 x 6 x 5 x 5) = 21600_[10] = 24 x 900
346236_[7] = 33 x 10362 = (3 + 4 + 6 + 2 + 3 + 6) x (3 x 4 x 6 x 2 x 3 x 6) = 62208_[10] = 24 x 2592
424644_[7] = 33 x 11646 = (4 + 2 + 4 + 6 + 4 + 4) x (4 x 2 x 4 x 6 x 4 x 4) = 73728_[10] = 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):