Simple Sexy Prime Pairs That are Happy Fathers Valid?

[Harry Neel]

Greetings To All:

While going from one place to another I started looking at 'happy' numbers that were also prime and that when their decimal digits were squared and then summed produced a second prime number (a father prime.) [This squaring of digits and then summing identified both happy numbers and potential father primes in one process.] 

In the process of this exercise it was noted that occasionally there were sexy prime pairs that were happy and also produced a father prime (or a child prime.)  For example:

     Prime 12143 goes to 31 and 12149 goes to 103, and 

     Prime 14321 goes to 31 and 14327 goes to 79.

After examining the natural numbers through prime number 32533, only seven (7) pairs of sexy primes have been identified that are also happy numbers and prime fathers. No pair has produced more than the father (the original prime number) and a child (the produced, or second prime number.) 

It is not known if the 'sexy happy father prime' pairs are finite, or not. (Dangerous extrapolation using twins, prime triplets, sexy prime triplets, etc., could indicate these are infinite, but not necessarily,)

The lessor terms (values, primes, etc.) of the seven (7) identified pairings are:

     12143, 14321, 21143, 21341, 23893, 30097, 32303.

To date, the values shown here have not been identified as a part of an OEIS sequence.

The primary question; is this anything that has a legitimate place as a sequence in OEIS?

Thanks to all for their consideration.

 

Sincerely,

Harry Neel

[Arthur O'Dwyer]

That is, you're asking about pairs (x,y) such that

- (x,y is a sexy pair of primes: x belongs to A023201 and y is equal to x+6) and also

- (x is happy: x belongs to A007770) and also

- (y is happy: y belongs to A007770) and also

- (the child of x is the lower of a sexy pair of primes: A003132(x) belongs to A023201) and also

- (the child of y is the upper of a sexy pair of primes:  A003132(y) belongs to A046117).

This doesn't seem to me like a very exciting set of numbers. Is there anything interesting you have to say about them, or anything you think anyone else might find interesting about them?

You said you checked up to 32533; but you can easily check up through 554893 using nothing more than https://oeis.org/A023201/b023201.txt .

–Arthur

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[Harry Neel]

Arthur,

Thanks for your comments and the links you provided.

The values are indeed extremely miniscule subsets of the natural numbers, primes, sexy primes, sexy prime triplets, happy primes, father primes, and more; including twins, cousins, triplets, etc..

Of interest to others. That is always the question, isn't it? Considering the relatively large number of consecutive primes that are six apart there are few pairs that are also happy and father primes. There are plenty of primes that are father primes that are not happy primes. There are also happy numbers that are not primes that, when digits are squared and summed, produce primes. These by definition are not considered to be father primes because they are not prime.

As an aside, while not evaluated to date, there are a number of child primes that are also Emirps. The 'offspring' of two pair of the sexy happy father primes are Emirps.

Perhaps I could have presented the query better. Maybe added a suggested definition, such as;

    Prime numbers k and k+6 that when their decimal digits are squared and then summed, the resulting value is another prime number which upon additional iterations ends in the number one (1). [Additional consecutive primes may be created in the process.]

Too long I know. At this juncture it is but an idea.

Harry

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