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):

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 |