Gapful Numbers Revisited

It was only recently (18th December 2024) that I made a post titled Gapful Numbers but today requires a second post about them because of the number associated with my diurnal age: 27740. This number marks the beginning of a run of three consecutive numbers with the properties that:

• the number is gapful, meaning that the number formed by concatenating the first and last digits of the number, divides the numbers
• the sum of the digits (SOD) of the number is also equal to the number formed by concatenating the first and last digits of the number

Thus we have:

• 27740 $\rightarrow$ 20 which divides it evenly and is equal to its SOD
• 27741 $\rightarrow$ 21 which divides it evenly and is equal to its SOD
• 27742 $\rightarrow$ 22 which divides it evenly and is equal to its SOD

In the range up to 40000, there are five groups of such triplets and they are:

• 10094,10095 and 10096
• 12255, 12256 and 12257
• 12256, 12257 and 12258
• 15134, 15135 and 15136
• 27740, 27741 and 27742

However, as can seen there is actually a run of four consecutive numbers in the above list, namely 12255, 12256, 12257 and 12258. While such runs of four are rare, they become more frequent if we only require the numbers to be gapful and not have the SODs equal to the concatenated first and last digits of the number. In the range up to 40000, the quadruplets are then (permalink):

• 10932, 10933, 10934 and 10935
• 11229, 11230, 11231 and 11232
• 12255, 12256, 12257 and 12258
• 15408, 15409, 15410 and 15411
• 16392, 16393, 16394 and 16395
• 17170, 17171, 17172 and 17173