Properties of Concatenated Numbers

Let's split a number in two parts and then consider the relationship that these two parts bear to the number of which both are a part. Now I've done this before as described in a post titled Energetic Numbers. In this post, I looked at numbers that belong to OEIS  A055480:

A055480

Energetic numbers: numbers that can be broken into two or more substrings and expressed as a sum of (possibly different) positive powers of those substrings.

This time I'll be looking at two different types of relationships. Let's consider 27573 and break it into two parts: 275 and 73. It turns out that my diurnal age today (27573) has an interesting property and it involves the totient function, commonly referred to as Euler's phi function or simply the phi function. We find that:$$ \begin{align}  \phi(27573) &= 1440 \\ \phi(275) \times \phi(73) &= 200 \times 72\\ &=1440 \end{align} $$Numbers like this belong to OEIS A147619 :

 A147619             Numbers \(n\)  = \(a \,| \, b\) such that \( \phi(n) = \phi(a) \times \phi(b) \)
                                where the symbol | represents concatenation

The initial members of this sequence are:

78, 780, 897, 918, 1179, 1365, 1776, 2574, 2598, 2967, 3168, 3762, 4758, 5775, 5796, 7800, 7875, 7917, 8217, 8970, 9180, 9576, 11790, 13650, 13662, 13875, 13896, 14391, 17760, 18564, 18858, 19812, 20097, 25740, 25935, 25974, 25980, 27573, 28776, 28779, 29670, 31680, 33165, 35919, 37620

Let's look at the breakdown for the numbers between 27573 and 37620:

• 275 | 73  -->  27573
  
  
•  287 | 76  -->  28776
  
  
•  287 |  79  -->  28779
  
  
•  29 | 670  -->   29670
  
  
•  31 | 680  -->   31680
  
  
•  331 | 65  -->   33165
  
  
•  35 | 919  -->   35919
  
  
•  37 | 620  -->   37620

A similar thing can be done with the sigma function that tallies the sum of the divisors of a number. Once again we break the number into two parts and then use the sigma function to connect those two parts to the whole. There is no OEIS function this time around but the property of the numbers being considered is as follows:

 Numbers \(n\)  = \(a \,| \, b\) such that \( \sigma(n) = \sigma(a) \times \sigma(b) \)
where the symbol | represent concatenation

The 78 numbers satisfying this condition between 27000 and 40000 are (permalink):

27175, 27298, 27418, 27445, 27500, 28018, 28195, 28750, 28798, 28978, 29038, 29058, 29098, 29278, 29395, 29398, 29500, 29875, 29875, 29950, 29950, 29980, 30498, 30775, 30788, 30989, 31750, 31795, 31918, 32595, 33175, 33238, 33298, 33725, 34189, 34555, 34557, 34795, 34975, 35338, 35395, 35578, 35585, 35670, 35818, 35975, 36178, 36238, 36350, 36398, 36426, 36775, 37138, 37241, 37678, 37750, 37798, 38247, 38370, 38518, 38570, 38638, 38750, 38826, 38856, 38877, 38940, 38975, 39118, 39425, 39478, 39500, 39754, 39805, 39898, 39976, 39980, 39992

Table 1 shows the breakdown for numbers between 27000 and 30000:

Table 1

Let's look at the first number 27175 where we break the number into 271 and 75 and we have:$$ \begin{align} \sigma(27175) &= 33728\\ \sigma(271) \times \sigma(75) &= 272 \times 124\\ &= 33728 \end{align}$$There are other possible concatenations involving more than two parts or other functions and properties of numbers so this post is merely a sample.