Thinning the Ranks

Every now and again I encounter a number associated with my diurnal age that seems to have no interesting properties from my perspective. In such cases, I have to be a little creative and such is the case for the number: 27355. After a little thought, I experimented with home primes and asked the question: how many iterations of factorise and concatenate are required to reach a home prime. Well, it turns out that three iterations are required:$$ \begin{align} 27355 &= 5 \times 5471 \rightarrow 55471 \\ 55471 &=13 \times 17 \times 251 \rightarrow 1317251 \\1317251 &=13 \times 19 \times 5333 \rightarrow 13195333 \end{align}$$Now in the range up to 40,000, there are well over 3,000 composite numbers with this property so it's hardly very special. These numbers belong to OEIS A046423:

A046423

Numbers requiring 3 steps to reach a prime under the prime factor concatenation procedure.

However, I noticed that the home prime for 27355 had exactly half of its digits equal to 3. It occurred to me to investigate how many composite numbers in the range up to 40,000 and belonging to OEIS A046423 had at least half their digits equal to 3. It turned out that there were only 191 numbers (permalink). This is what I meant by "thinning the ranks". It's interesting to investigate the frequency for other digits that comprise at least half of the digits of the home prime. Here are the statistics:

• 0 --> no numbers

• 1 --> 120 numbers

• 2 --> 2 numbers

• 3 --> 191 numbers

• 4 --> 3 numbers

• 5 --> 2 numbers

• 6 --> no numbers

• 7 --> 69 numbers

• 8 --> no numbers

• 9 --> 11 numbers

The algorithm is easily modified to accommodate different size iterations. Here are some more statistics for different numbers of iterations involving the digit 3 in the range up to 40,000 (permalink):

• 1 iteration --> 600 numbers

• 2 iterations --> 441 numbers

• 3 iterations --> 191 numbers

• 4 iterations --> 105 numbers

• 5 iterations --> 48 numbers

• 6 iterations --> 34 numbers

• 7 iterations --> 12 numbers

• 8 iterations --> 6 numbers

• 9 iterations --> no numbers

So as can be seen, any number can be made more special by imposing more conditions. For example, the numbers that require 8 iterations to reach a home prime that has at least half of its digits equal to 3 are 4017, 4242, 4667, 7474, 31355 and 39309 with the following details (permalink):

• 4017 --> 3337715393

• 4242 --> 23393307373

• 4667 --> 33433193

• 7474 --> 23393307373

• 31355 --> 3332943503

• 39309 --> 3337715393

The very next number after 27355 provides another excellent opportunity to thin the ranks. This is because 27356 is only one step removed from its home prime:$$27356 \rightarrow 227977$$Now 16.8% of numbers in the range up to 40,000 have this property so again its hardly special. However, 27356 has the property that its home prime contains three occurrences of the digit "7". Only 139 or 0.348 % of the numbers have this property (see permalink). Again the algorithm is easily modified to accommodate other digits. Here are the statistics:

• digit 0 occurs 3 times --> no numbers

• digit 1 occurs 3 times --> 440 numbers

• digit 2 occurs 3 times --> 645 numbers

• digit 3 occurs 3 times --> 565 numbers

• digit 4 occurs 3 times --> 4 numbers

• digit 5 occurs 3 times --> 83 numbers

• digit 6 occurs 3 times --> 1 number

• digit 7 occurs 3 times --> 139 numbers

• digit 8 occurs 3 times --> 1 number

• digit 9 occurs 3 times --> 23 numbers

Yes another example is provided by OEIS A187073:

A187073

Composite square-free numbers whose average prime factor is a prime number.

In the range up to 40,000, there are 1609 such numbers. How can we thin this sequence out? Well, firstly let's consider only sphenic numbers. This immediately cuts the number to 594. Now how many of these follow a 1-2-3 progression in terms of the lengths of their prime factors? Only 308. Still too many? Let's apply the condition that the prime factors can have no digits in common. This leaves us with only 13 and these numbers are:

• 15369 = 3 * 47 * 109 with prime average of 53
• 15515 = 5 * 29 * 107 with prime average of 47
• 17135 = 5 * 23 * 149 with prime average of 59
• 22865 = 5 * 17 * 269 with prime average of 97
• 24215 = 5 * 29 * 167 with prime average of 67
• 26619 = 3 * 19 * 467 with prime average of 163
• 29949 = 3 * 67 * 149 with prime average of 73
• 32809 = 7 * 43 * 109 with prime average of 53
• 33065 = 5 * 17 * 389 with prime average of 137
• 33909 = 3 * 89 * 127 with prime average of 73
• 36879 = 3 * 19 * 647 with prime average of 223
• 37639 = 7 * 19 * 283 with prime average of 103
• 39759 = 3 * 29 * 457 with prime average of 163

Notice that all these numbers, while conforming to the requirement of OEIS A187073, are sphenic, have factors in a 1-2-3 progression with no digits in common between any of the factors.