It was only after visiting https://factordb.com, that I noticed there was an option for calculating inverse home primes in various bases. This is the site that enabled me to view the aliquot sequence for 27528 up to 2338 steps which I reported on in my previous post titled Infinite, Aperiodic Aliquot Sequences. The concept of inverse home primes is quite simple really and involves the catcatenation of the prime factors of a number from highest to lowest rather than from lowest to highest as is the case for home primes. I made a post recently titled Numbers With Elusive Home Primes.
Let's illustrate the difference between home primes and inverse home primes using 875 as an example.
HOME PRIME
\(875 = 5^3 \times 7\)
\(5557 = 5557\) this is step 1
The number of steps required is to reach home prime is 1 : [875, 5557]
INVERSE HOME PRIME
The number of steps required is to reach an inverse home prime is 8 :\(875 = 5^3 \times 7\)
\(7555 = 5 \times 1511\) this is step 1
\(15115 = 5 \times 3023\) this is step 2
\(30235 = 5 \times 6047\) this is step 3
\(60475 = 5^2 \times 41 \times 59\) this is step 4
\(594155 = 5 \times 118831\) this is step 5
\(1188315 = 3^2 \times 5 \times 26407\) this is step 6
\(26407533 = 3 \times 8802511\) this is step 7
\(88025113 = 11 \times 8002283\) this is step 8
\(800228311 = 800228311\) this is step 9
[7555, 15115, 30235, 60475, 594155, 1188315, 26407533, 88025113, 800228311]
... what a big difference the order makes when concatenating the factors of a number ...
However, it turns out that inverse home primes are far more "elusive" than the familiar home primes. This is not surprising. Any even number will have the digit 2 as its smallest factor which means that it will be the LAST digit of the newly concatenated number and thus even again. This means that any even number CANNOT have an inverse home prime because of this endless cycle.
Similarly a number ending in the digit 5 is likely to have this digit as the LAST digit of the newly concatenated number and thus it will not be prime but divisible by 5 again. I say "likely" because numbers ending in 5 can escape an endless cycle once 3 becomes a factor. We see this in the example of 875 shown above. Numbers ending in 1, 3, 7 and 9 are likely to fare much better in finding their inverse home primes than numbers ending in 5 and for numbers ending in 0, 2, 4, 6 and 8 inverse home primes are simply not possible.
The question arises however, as to whether even numbers can enter a cycle during the concatentation process. Take a number as simple as 22. After 47 steps, we have a 127 digit number:
494775798475134788874946695481884739784434386126997999116887215921268574642953570240011863034391409057404851408757947611917322
So cycles seem unlikely. There's nothing about inverse home primes on the Internet except for the site mentioned earlier. However, it's interesting to realise what a big difference the order makes when concatenating the factors of a number. I've incorporated the calculation of inverse home primes into my multipurpose algorithm just to explore them in greater depth.
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