Sum Product Numbers and Near Misses

A sum-product number is a number \(n\) such that the sum of its digits times the product of its digit is \(n\) itself. For example:$$135=(1+3+5)\times (1 \times 3 \times 5)$$There are only three sum-product numbers: 1, 135, and 144 (OEIS A038369), although 0 of course could be included if desired.

A038369

Numbers n such that n = (product of digits of n) * (sum of digits of n).

Of interest also are numbers such that the sum of their digits times the product of their digit differ from by 1, 2, 3, … The table in Figure 1 shows these initial numbers (source):

Figure 1

The code to determine these initial numbers is shown in Figure 2 with a difference of 6 being used as the example:

Figure 2: permalink

Of course, if the number base is changed (say to 16), an entirely different set of numbers is obtained. In base 16 and with a number difference of 6, the set consists of 3, 482, 554, 582, 7494, 12954. In the case of 12954, it becomes 329A and so the sum of its digits is 3 + 2 + 9 + 10 = 24 and the product of its digits is 3 x 2 x 9 x 10 = 540. The product of 24 and 540 is 12960 which does indeed differ from 12954 by 6. See Figure 3.

Figure 3: permalink

These numbers, in base 10 or any other base, are curiosities and probably, like selfie numbers, not of any deep mathematical significance. However, I did turn 25926 days old recently (March 27th 2020) and that's when I stumbled upon the number thanks to a Wolfram MathWorld link. It features in the set of numbers shown in Figure 2 and it also appears in the table in Figure 1.