Some Special Pronic Numbers

While driving I'm always on the lookout for interesting number plates and this morning, while driving, I noticed a number plate with numerals that were a concatenation of two consecutive numbers. I can't remember what the number was but the thought then struck me as to whether there were any pronic numbers that could be constructed from concatenated consecutive numbers. If there were then the number would have the property that it was a product of consecutive numbers as well as a concatenation of consecutive numbers (not necessarily the same numbers however).

It's easy enough to check for (permalink) and in the range up to one hundred million, there are only three numbers that qualify. These are 12, 56 and 6162:

• \(12 = 1 \, || \,  2 =2^2 \times 3 = 3 \times 4 \)
• \(56 = 5 \, || \, 6 =2^3 \times 7=7 \times 8 \)
• \(6162 = 61 \, || \, 62 = 2 \times 3 \times 13 \times 79 = 78 \times 79 \)

This stimulated me to look for some other special pronic numbers. My thoughts turned to numbers that share the same digits as their pronic factors. In the range up to slightly over one hundred million, there are quite a few. They are (permalink):

110, 10100, 11990, 123552, 251502, 1001000, 1019090, 1156700, 2987712, 11232552, 13701102, 25015002, 25255650, 25265702, 25720112, 57010050, 57161160, 100010000

The details are as follows:

• 110 = 10 x 11 with shared digits of 0, 1
• 10100 = 100 x 101 with shared digits of 0, 1
• 11990 = 109 x 110 with shared digits of 0, 1, 9
• 123552 = 351 x 352 with shared digits of 1, 2, 3, 5
• 251502 = 501 x 502 with shared digits of 0, 1, 2, 5
• 1001000 = 1000 x 1001 with shared digits of 0, 1
• 1019090 = 1009 x 1010 with shared digits of 0, 1, 9
• 1156700 = 1075 x 1076 with shared digits of 0, 1, 5, 6, 7
• 2987712 = 1728 x 1729 with shared digits of 1, 2, 7, 8, 9
• 11232552 = 3351 x 3352 with shared digits of 1, 2, 3, 5
• 13701102 = 3701 x 3702 with shared digits of 0, 1, 2, 3, 7
• 25015002 = 5001 x 5002 with shared digits of 0, 1, 2, 5
• 25255650 = 5025 x 5026 with shared digits of 0, 2, 5, 6
• 25265702 = 5026 x 5027 with shared digits of 0, 2, 5, 6, 7
• 25720112 = 5071 x 5072 with shared digits of 0, 1, 2, 5, 7
• 57010050 = 7550 x 7551 with shared digits of 0, 1, 5, 7
• 57161160 = 7560 x 7561 with shared digits of 0, 1, 5, 6, 7
• 100010000 = 10000 x 10001 with shared digits of 0, 1

There's plenty more room for investigation but I'll leave if there for now.