Numbers Within Numbers

Let's formalise the concept of internal digits. Take a number like 28211. It's prime but we remove the leftmost digits (2) and the rightmost digit (1), we are left with the number 821. This the number within a number and it happens to be also prime. It is thus a member of OEIS A069686:

 A069686: primes whose internal digits form a prime.

The members of this sequence from 28000 to 40000 are (permalink):

28097, 28099, 28111, 28211, 28219, 28277, 28279, 28297, 28393, 28537, 28571, 28573, 28579, 28591, 28597, 28631, 28771, 28813, 28817, 28837, 28871, 28879, 29077, 29191, 29297, 29411, 29473, 29531, 29537, 29671, 29717, 29833, 29837, 29917, 30029, 30059, 30071, 30113, 30119, 30133, 30137, 30139, 30197, 30293, 30313, 30319, 30431, 30539, 30593, 30671, 30677, 30713, 30839, 30893, 30971, 30977, 31013, 31019, 31033, 31039, 31079, 31091, 31139, 31271, 31277, 31319, 31379, 31391, 31393, 31397, 31511, 31513, 31517, 31573, 31793, 31799, 31817, 31973, 31991, 32117, 32119, 32233, 32237, 32297, 32299, 32411, 32413, 32573, 32579, 32633, 32693, 32713, 32717, 32719, 32771, 32779, 32831, 32833, 32839, 32933, 32939, 33071, 33073, 33113, 33119, 33179, 33311, 33317, 33377, 33479, 33493, 33533, 33599, 33679, 33739, 33791, 33797, 33893, 34019, 34211, 34213, 34217, 34313, 34319, 34337, 34439, 34499, 34613, 34631, 34673, 34679, 34871, 34877, 34913, 34919, 35099, 35419, 35573, 35771, 35879, 35933, 35993, 35999, 36011, 36013, 36017, 36073, 36131, 36137, 36191, 36313, 36319, 36433, 36473, 36479, 36599, 36739, 36779, 36833, 36913, 36919, 37013, 37019, 37097, 37199, 37273, 37277, 37337, 37339, 37397, 37511, 37517, 37571, 37573, 37579, 37619, 37691, 37693, 37699, 37871, 37879, 38113, 38119, 38219, 38231, 38237, 38239, 38273, 38299, 38393, 38593, 38639, 38833, 38839, 38873, 39079, 39113, 39119, 39191, 39199, 39293, 39371, 39373, 39419, 39671, 39679, 39719, 39779, 39839, 39971, 39979

We can modify the algorithm to search for square numbers within square numbers. For example, \(144 = 12^2\) and its internal digit \(4 = 2^2\) and so it is a member of OEIS A069701:

 A069701: squares with internal digits also forming a square > 0.

The members up to 40000 are few and far between: 

144, 196, 441, 841, 1369, 3364, 4096, 5041, 8649, 10816, 11449, 20164, 38416.

So the takeaway from this is that every \(n\) digit number has an (\(n-2\)) digit number inside of it provided \(n>2\) and sequences can be developed by relating this internal number to the larger number of which it forms a part.

Let's consider numbers that are biprime and whose internal digits also form a biprime. Take 28189 as an example:$$ \begin{align} 28189 &= 7 \times 4027 \\ 818 &= 2 \times 409 \end{align}$$These sorts of numbers are not listed in the OEIS. 

Similarly for triprimes. Take 28055 as an example:$$ \begin{align} 28055 &= 5 \times 31 \times  181\\805 &= 5 \times 7 \times 23 \end{align}$$Of course we could consider numbers within numbers within numbers etc. but given that I'm mainly examining five digit numbers less than 40000, I'll leave off at numbers within numbers. Of course, these relationships between numbers and their respective internal numbers only apply with certainty in the number base under consideration (base 10 so far in this post). Take our earlier example of 28211 and its internal number of 821. Both are prime. Let's change to base 9:$$ \begin{align} 28211_{10} &= 42625_9 \rightarrow 265_9 \text{ as its internal number} \\ 265_9 &= 218_{10} \text{ which is clearly not prime} \end{align} $$Primeness is base independent and if a number is not prime in a certain base then it is not prime in any base. However, the inner number while not prime in base 10 may be prime if regarded as a number in another base. Let's illustrate this with 28019. It is a prime in base 10 where the inner number, 801, is divisible by 3. However, we can note the following:$$ \begin{align} 801_{12} &= 1153_{10} \text{ which is prime}\\801_{15} &= 1801_{10} \text{ which is prime} \end{align} $$I've incorporated the search for the following types of numbers into my number analysis algorithm so that the following types of numbers can be identified:

• prime number whose internal number is also prime in any base from 2 to 16
• biprime number whose internal number is also prime using base 10 only
• triprime number whose internal number is also triprime using base 10 only
• square number whose internal number is also square using base 10 only