[ April 20, 2015 ]
A new twist on Home Primes
Jeff Heleen (email)

It's called Doubly Prime Home Primes (DPHP).

Jeff Heleen poured his data also in a convenient
OEIS sequence see A257496

[ If you're not yet familiar with Home Primes please visit
first my webpage Home Primes ]

To get one, just take all the factors generated during the search
for a home prime, put them together in the order found (just like
when looking for the home prime itself) and test for primality. If
it is a prime you have a DPHP.

Ex. Starting number 16 takes 4 iterations to get to the home prime
of 31636373.

\(\bbox[13px,border:1px solid green]{ \begin{align} {\color{blue}{16}} &\to 2 * 2 * 2 * 2\\ 2222 &\to 2 * 11 * 101\\ 211101 &\to 3 * 11 * 6397\\ 3116397 &\to 3 * 163 * 6373\\ 31636373 &\to \text{is prime} \end{align} }\)

Now take all the factors found, in order:
2222211101311639731636373 prime.

So 16 leads to a DPHP.

The number of iterations must be more than 1, preventing
numbers such as 12 (12 → 2 * 2 * 3 → 223) from being a DPHP.

An additional restriction is that a number is not a DPHP if
it appears as the result of a product in any home prime search.
So 25 is not a DPHP because
a(10) => 2 * 5 → 25 5 * 11 → 511 7 * 73 → 773,
and 25 appears in the process.
The start number must be unique, ie, it can't be an internal member
of another chain of numbers. Another way of saying it is you can't be able
to generate it from another number. It is the entire chain, not a partial
chain, that is checked for primality.

I have checked all unique start numbers smaller and equal to 100 (except 49)
and the following generate DPHP's: 9, 16, 26, 42, 74, 75 and 95.

These DPHP's even contain an example of 2 unique start numbers
that generate the same HP and DPHP, 42 and 74 through different
paths: 42 = 2 * 3 * 7 and 74 = 2 * 37.