More Numbers as Concatenations

In my previous post, Divisors and Antidivisors: A Fresh Perspective, I dealt with numbers formed by concatenations of divisors and also by antidivisors of a number. For example, consider the number 15:

• 15 has divisors of 1, 3, 5 and 15
• 13515 is formed by concatenating these divisors
• 15 divides 13515 to give 901
• numbers with this property belong to OEIS A069872
• 15 has proper divisors of 1, 3 and 5
• 135 is formed by concatenating these proper divisors
• 15 divides 135 to give 9
• numbers with this property belong to OEIS A240265
• 15 has antidivisors of 2, 6 and 10
• 2610 is formed by concatenating these antidivisors
• 15 divides 2610 to give 174
• numbers with this property belong to OEIS A249764

In an earlier post titled, Nothing New Under The Sun, I looked at numbers formed by concatenation of powers of prime digits. Let's take 128864 which can be formed by a concatenation of powers of 2:$$128864=2^7 \, | \, 2^3 \, | \, 2^6$$where | is the symbol commonly used for concatenation. Numbers like this belong to OEIS A381259. 

In this post I want to look at numbers that are a concatenation of the multiples of a digit but that do not contain the digit itself. Let's take 28133 as an example. It's not immediately obvious but this number can be broke into two parts, 28 and 133, both of which are multiples of 7:$$28133 \rightarrow 28 \, | \, 133 = (7 \times 4) \, | \, (7 \times 19)$$There are 190 such numbers in the range up 40000. They are (permalink):

1414, 1421, 1428, 1435, 1442, 1449, 1456, 1463, 1484, 1491, 1498, 2121, 2128, 2135, 2142, 2149, 2156, 2163, 2184, 2191, 2198, 2828, 2835, 2842, 2849, 2856, 2863, 2884, 2891, 2898, 3535, 3542, 3549, 3556, 3563, 3584, 3591, 3598, 4242, 4249, 4256, 4263, 4284, 4291, 4298, 4949, 4956, 4963, 4984, 4991, 4998, 5656, 5663, 5684, 5691, 5698, 6363, 6384, 6391, 6398, 8484, 8491, 8498, 9191, 9198, 9898, 14105, 14112, 14119, 14126, 14133, 14140, 14154, 14161, 14168, 14182, 14189, 14196, 14203, 14210, 14224, 14231, 14238, 14245, 14252, 14259, 14266, 14280, 14294, 14301, 14308, 14315, 14322, 14329, 14336, 14343, 14350, 21105, 21112, 21119, 21126, 21133, 21140, 21154, 21161, 21168, 21182, 21189, 21196, 21203, 21210, 21224, 21231, 21238, 21245, 21252, 21259, 21266, 21280, 21294, 21301, 21308, 21315, 21322, 21329, 21336, 21343, 21350, 28105, 28112, 28119, 28126, 28133, 28140, 28154, 28161, 28168, 28182, 28189, 28196, 28203, 28210, 28224, 28231, 28238, 28245, 28252, 28259, 28266, 28280, 28294, 28301, 28308, 28315, 28322, 28329, 28336, 28343, 28350, 35105, 35112, 35119, 35126, 35133, 35140, 35154, 35161, 35168, 35182, 35189, 35196, 35203, 35210, 35224, 35231, 35238, 35245, 35252, 35259, 35266, 35280, 35294, 35301, 35308, 35315, 35322, 35329, 35336, 35343, 35350

Choosing multiples of 2 and 3 produce 641 and 455 suitable numbers respectively while choosing multiples of 5 produces 78 suitable numbers in the range up to 40000 (permalink):

1010, 1020, 1030, 1040, 1060, 1070, 1080, 1090, 2020, 2030, 2040, 2060, 2070, 2080, 2090, 3030, 3040, 3060, 3070, 3080, 3090, 4040, 4060, 4070, 4080, 4090, 6060, 6070, 6080, 6090, 7070, 7080, 7090, 8080, 8090, 9090, 10100, 10110, 10120, 10130, 10140, 10160, 10170, 10180, 10190, 10200, 10210, 10220, 10230, 10240, 20100, 20110, 20120, 20130, 20140, 20160, 20170, 20180, 20190, 20200, 20210, 20220, 20230, 20240, 30100, 30110, 30120, 30130, 30140, 30160, 30170, 30180, 30190, 30200, 30210, 30220, 30230, 30240

In the case of 1010, we have:$$1010 \rightarrow 10 \, | \, 10 = (5 \times 2) \, | \, (5 \times 2)$$The permalink allows experimentation with other digits or even numbers. There's no deep Mathematics in all this just another way to spot patterns in numbers.