Claude to the Rescue

The number 27983, associated with my diurnal age today, has a property than confers it membership in OEIS A335752: the lexicographically earliest sequence of primes, each two of which sum to a triprime. The sequence begins: 3, 5, 47, 239, 3023, 27983, 53231, ... so let's confirm that these numbers satisfy using 3 as a starting point:

• \(3 + 5 = 8 =2^3\)
• \(3 + 47 = 50 = 2 \times 5^2\)
• \(3 + 239 = 242 = 2 \times 11^2 \)
• \(3 + 3023 = 3026 = 2 \times 17 \times 89 \)
• \(3+27983 = 27986 = 2 \times 7 \times 1999\)
  
  

The sums above are all triprimes and we will find that any number in the list paired with any other number will produce a triprime. The code for this was included in the OEIS comments but was in Mathematica. I got Gemini to convert it to Python code and then Grok but in both cases the primeomega function was misunderstood with both claiming that primeomega(12) =2 and not 3. 

I didn't fully understand the Python code and was looking to Gemini and then Grok to explain it to me. It was only Claude that succeeded to my satisfaction. Here is the code followed by Claude's explanation. I think Claude will be my go to resource for anything to do with coding.

from sympy import nextprime, primeomega

ps = [3]

for _ in range(10):

    p = nextprime(ps[-1]) 

    while not all(primeomega(x + p) == 3 for x in ps):

        p = nextprime(p)

    ps.append(p)

print(ps)

I'd be happy to explain this code line by line! This code generates a sequence of prime numbers with a special property.

Code Breakdown

from sympy import nextprime, primeomega

Imports two functions from SymPy:

• nextprime(\(n\)): Returns the next prime number greater than \(n\)
• primeomega(\(n\)): Returns the total number of prime factors of \(n\) (counting multiplicity). For example, primeomega(12) = 3 because 12 = 2² × 3 (two 2's and one 3)

ps = [3]

Initializes the sequence with the first prime number 3. This list will store our special sequence of primes.

for _ in range(10):

Loops 10 times to find 10 more primes (so we'll have 11 total including the initial 3). The underscore _ is used when we don't need the loop variable.

p = nextprime(ps[-1])

Gets the next prime after the last element in our list. 

• ps[-1] accesses the last element of the list
• Initially, this gives nextprime(3) = 5
• This is our candidate prime to add to the sequence

while not all(primeomega(x + p) == 3 for x in ps):

    p = nextprime(p)

This is the key part! It checks if the candidate prime \(p\) satisfies a special condition:

• For every number \(x\) already in our list ps
• The sum \(x + p\) must have exactly 3 prime factors (counting multiplicity)
• If this condition fails for any \(x\), we move to the next prime candidate
• This continues until we find a prime where all sums have exactly 3 prime factors

ps.append(p)

Adds the qualified prime to our sequence once it passes the test.

print(ps)

Prints the final sequence** of primes.

Example Walkthrough

Let's trace the first iteration:

1. Start with ps = [3]

2. Try p = 5: Check if 3 + 5 = 8 = 2³ has 3 prime factors ✓ (yes, three 2's)

3. Add 5 to list: ps = [3, 5]

4. Try p = 7: Check 3 + 7 = 10 = 2 × 5 (only 2 factors) ✗

5. Try p = 11: Check 3 + 11 = 14 = 2 × 7 (only 2 factors) ✗

6. Continue until finding a prime that works with both 3 and 5...

This creates a sequence where each new prime maintains the property that when added to any previous prime in the sequence, the result always has exactly 3 prime factors!

*************************

Now if we change the condition to primeomega = 2, we get the following sequence: 3, 7, 19, 139, 859, 8179, 173059, ... where it can be seen there is a big jump from 8179 to 173059. Testing this out using 3 again we see that:

• \(3+7=10=2 \times 5\)
• \(3+19=22=2 \times 11\)
• \(3+139=142=2 \times 71\)
• \(3+859=862=2 \times 431\)
• \(3+8179=8182=2 \times 4091\)
• \(3+173059=173062=2 \times 86531\)

The sequence is included in the OEIS as A113875: slowest growing sequence of primes having the prime-pairwise-average property: if i<j, (a(i)+a(j))/2 is prime. This makes sense in that 2 seems to be the common factor is all of the sums. 

The sequence with additional (very large) terms looks like this:

3, 7, 19, 139, 859, 8179, 173059, 1026199, 1827139, 15828679, 13187242759, 18732483199, 912492556939, 9130567625119

When we set primeomega = 4 we get the following sequence: 3, 13, 137, 1193, 4373, 26017, 33889, 495433, ... which is NOT listed in the OEIS. Again let's use 3 as our test number:

• \(3+13=16=2^4\)
• \(3+137=140=2^2 \times 5 \times 7\)
• \(3+1193=1196=2^2 \times 13 \times 23\)
• \(3+4373=4376=2^3 \times 547\)
• \(3+26017=26020=2^2 \times 5 \times 1301\)
• \(3+33889=33892=2^2 \times 37 \times 229\)
• \(3+495433=495436=2^2 \times 29 \times 4271\)

That's probably a good place to stop.

Building Sequences from a Seed Pair

 FIRST EXAMPLE

The numbers 1 and 4 have the interesting properties that:

• their sum is prime: 1 + 4 = 5
• their difference is prime: 4 - 1 = 3
• their product (4) is the average of a pair of twin primes (3 and 5)

Let's make this the starting point of a sequence and let the third member of the sequence be \(x\). This gives us: \(1, 4, x\). We want 4 and \(x\) to share the properties that 1 and 4 enjoyed. Namely:

• 4 + \(x\) is prime
• \(x\) - 4 is prime
• 4 \(\times x\) is the average of a pair of twin primes

A little trial and error shows that the smallest value of \(x\) we are looking for is 15 because:

• 4 + 15 = 19 is prime
• 15 - 4 = 11 is prime
• 4 \( \times \) 15 = 60 is average of a pair of twin pairs (59 and 61)

By using not trial and error but a simple algorithm we can find further terms. The result is OEIS A154493 and the initial terms are:

1, 4, 15, 28, 39, 50, 81, 350, 459, 512, 675, 944, 987, 1040, 1917, 1936, 2325, 2378, 2421, 2588, 2745, 2812, 3459, 3488, 3495, 3506, 5667, 5804, 6027, 6074, 24765, 24832, 25479, 25552, 27621, 27848, 27951, 27980, 34101, 34720, 34773, 35344

SECOND EXAMPLE

Let's take another seed pair with the simple property that the two numbers must add to a cubic number. We'll use 1 and 7 as our seed pair because: \(1 + 7 = 8 = 2^3\). Here we see that the next number must be 20 because \(7+20=27=3^3\). Using an algorithm, the sequence generated begins thus (permalink):

1, 7, 20, 44, 81, 135, 208, 304, 425, 575, 756, 972, 1225, 1519, 1856, 2240, 2673, 3159, 3700, 4300, 4961, 5687, 6480, 7344, 8281, 9295, 10388, 11564, 12825, 14175, 15616, 17152, 18785, 20519, 22356, 24300, 26353, 28519, 30800, 33200, 35721, 38367

THIRD EXAMPLE

Let's start with seed numbers 1 and 2 this time with the property that:

• the sum of the two numbers has a digit sum that is prime
• the product of the two numbers plus 1 has a digit product that is prime

The seed pair 1 and 2 satisfy since:

• the sum of 1 and 2 is 3 and 3 is prime
• the product of 1 and 2 plus 1 is 3 and 3 is prime

This leads to the following sequence: 1, 2, 9, 12, 13, 16, 18, 23, 24, 25, 27, 29, 32, 33, 34, 40, 45, 47, 51, 60, 62, 66, 100, ... (permalink). There are 2211 terms in the range up to 40000.

Xenodrome Pairs, Triplets Etc.

\( \textbf{27815} \), the number associated with my diurnal age today, is one of those numbers for which an interesting property is hard to come by. However, I did notice that the number was a xenodrome in both base 10 and base 9 (42135). A little further investigation revealed that 27816 also shared this property. That got me thinking about how often pairs of such numbers occur. Now I've written about xendromes before in posts titled Xenodromes and Xenodrome Probabilities in which I've looked at numbers that remain xenodromes in various bases but so far I've not looked at groupings of numbers by pairs, triplets, quadruplets etc. with properties relating to xenodromes.

In my Bespoken for Sequences database, I've listed the following sequence of numbers:$$ \begin{align} \textbf{Smaller of a pair of consecutive numbers} \\ \textbf{that are xenodromes in base 10 and base 9} \end{align} $$Between 27815 and 40000, there are 282 numbers with this property (permalink):

27815, 27834, 27835, 27845, 27860, 27950, 27953, 27960, 28013, 28016, 28134, 28169, 28195, 28196, 28314, 28346, 28356, 28364, 28395, 28456, 28495, 28509, 28536, 28563, 28573, 28590, 28609, 28670, 28914, 28934, 28935, 28963, 29015, 29016, 29053, 29075, 29084, 29085, 29103, 29134, 29147, 30147, 30148, 30156, 30157, 30186, 30194, 30195, 30196, 30457, 30458, 30467, 30528, 30548, 30561, 30591, 30691, 30724, 30725, 30751, 30764, 30814, 30825, 30851, 30924, 30925, 31024, 31025, 31048, 31057, 31086, 31094, 31095, 31096, 31097, 31257, 31258, 31259, 31267, 31268, 31284, 31294, 31475, 31586, 31604, 31607, 31608, 31806, 31824, 31825, 31826, 31859, 31905, 31906, 31907, 32018, 32104, 32108, 32159, 32189, 32496, 32508, 32509, 32546, 32580, 32589, 32590, 32607, 32608, 32609, 32648, 32657, 32670, 32689, 32690, 32697, 32907, 32947, 32960, 32964, 34027, 34058, 34085, 34086, 34091, 34095, 34127, 34157, 34158, 34175, 34185, 34206, 34215, 34275, 34278, 34279, 34296, 34297, 34560, 34567, 34568, 34569, 34578, 34620, 34650, 34658, 34761, 34785, 34815, 34820, 34905, 34917, 34926, 35016, 35017, 35047, 35097, 35169, 35196, 35197, 35208, 35216, 35217, 35479, 35486, 35496, 35641, 35809, 35826, 35890, 35891, 35916, 35917, 35946, 35961, 35970, 35971, 35980, 35981, 36018, 36027, 36028, 36208, 36209, 36214, 36218, 36270, 36278, 36279, 36280, 36289, 36290, 37195, 37204, 37245, 37285, 37294, 37295, 37458, 37459, 37485, 37495, 37519, 37520, 37528, 37568, 37819, 37820, 37845, 37864, 37890, 37891, 37920, 37945, 37964, 37981, 38015, 38024, 38045, 38046, 38064, 38105, 38106, 38124, 38145, 38159, 38169, 38205, 38206, 38240, 38249, 38250, 38259, 38260, 38405, 38406, 38415, 38420, 38469, 38649, 38670, 38674, 38694, 38701, 38720, 38724, 38751, 38760, 38904, 38916, 39015, 39016, 39017, 39024, 39025, 39026, 39124, 39125, 39126, 39186, 39205, 39206, 39207, 39215, 39216, 39240, 39260, 39481, 39485, 39540, 39541, 39571, 39580, 39604, 39701, 39715, 39724, 39764, 39805, 39814, 39845, 39846

There are 69  triplets in the same range (permalink):$$ \begin{align} \textbf{Smallest of a triplet of consecutive numbers} \\ \textbf{that are xenodromes in base 10 and base 9} \end{align} $$27834, 28195, 28934, 29015, 29084, 30147, 30156, 30194, 30195, 30457, 30724, 30924, 31024, 31094, 31095, 31096, 31257, 31258, 31267, 31607, 31824, 31825, 31905, 31906, 32508, 32589, 32607, 32608, 32689, 34085, 34157, 34278, 34296, 34567, 34568, 35016, 35196, 35216, 35890, 35916, 35970, 35980, 36027, 36208, 36278, 36279, 36289, 37294, 37458, 37519, 37819, 37890, 38045, 38105, 38205, 38249, 38259, 38405, 39015, 39016, 39024, 39025, 39124, 39125, 39205, 39206, 39215, 39540, 39845

An example is the xenodrome 27834 with 27835 and 27836 also xenodromes. The base 9 equivalents 42156, 42157 and 42158 are also xenodromes.

There are 13 quadruplets in the same range (permalink):$$ \begin{align} \textbf{Smallest of a quaduplet of consecutive numbers} \\ \textbf{that are xenodromes in base 10 and base 9} \end{align} $$30194, 31094, 31095, 31257, 31824, 31905, 32607, 34567, 36278, 39015, 39024, 39124, 39205

An example is the xenodrome 30194 with 30195, 30196 and 30197 also xenodromes. The base 9 equivalents 45368, 45370, 45371 and 45372 are also xenodromes. 

There is only one \( \textbf{quintuplet}\) in the range and that is 31094. Here we see that 31094, 31095, 31096, 31097 and 31098 are all xenodromes as are their base 9 equivalents 46578, 46580, 46581, 46582 and 46583.

Numbers With Elusive Home Primes

I was surprised to find that the number associated with my diurnal age today, 27525, is a number for which a home prime is not able to be ascertained. See Figure 1 where the algorithm applied to this number quits after 50 concatenations fail to generate a prime.

Figure 1: permalink

However, what surprised me even further was that the very next number, 27526, is also a number for which a home prime is not able to be ascertained. Such pairs are not unprecedented, the first being 714 and 715 followed by 1138 and 1139. See my blog post Home Primes from 2nd May 2021. The very first number for which a home prime cannot be found is 49.

After 27526, the next number 27527 is prime and I've written about this number in a post What's Special About 27527 on 10th August 2024. After that, 27528 requires ten steps and 27529 is a prime. I'll continue to monitor the home primes for forthcoming numbers. There's no way of knowing whether these numbers for which home primes are currently unobtainable will not eventually be found to reach home primes once quantum computers are focused on the problem.

Divisibility Sequences

It's easy to miss. The square numbers are 1, 4, 9, 16, 25, 36, 49 and so on but it's not obvious that the consecutive integers 27423, 27424 and 27425 are divisible by consecutive square numbers. Thus we have:$$ \begin{align} 27423 &= 3^2 \cdot 11 \cdot 277 \text{ divisible by }9=3^2\\27424 &= 2^5 \cdot 857 \text{ divisible by }16=4^2\\27425 &= 5^2 \cdot 1097 \text{ divisible by }25=5^2 \end{align}$$I only noticed this fact because my diurnal age today is 27423 and this number is a member of OEIS A178919:

A178919

Smallest of three consecutive integers divisible respectively by three consecutive squares greater than 1.

Membership of this sequence does not come easy and can be seen in the list of its initial members (permalink):

2223, 5823, 9423, 13023, 16623, 20223, 23823, 27423, 31023, 32975, 34623, 38223, 41823, 45423, 49023, 52623, 56223, 59823, 63423, 67023, 70623, 74223, 77075, 77823, 81423, 85023, 88623, 92223, 95823, 99423, 103023, 106623, 110223

Not surprisingly membership in the equivalent sequence of two consecutive integers divisible by two consecutive squares is a lot easier. This sequence is OEIS A178918. The natural question to ask is whether there are groups of four consecutive integers divisible by four consecutive squares. Testing up in the range up to ten million, we find no such groups. However, they may well exist further out.

What about cubes? Can we find groups of three consecutive integers that are divisible by three consecutive cubes greater than 1. Indeed we can and, up one million, the sequence of the smallest members of these trios is (permalink):

106623, 322623, 538623, 754623, 970623 (not listed in the OEIS)

Let's look at the first member of the sequence where we find:$$\begin{align} 106623 &= 3^3 \cdot 11 \cdot 359 \text{ divisible by } 27 =3^3\\106624 &= 2^7 \cdot 7^2 \cdot 17 \text{ divisible by }64 =4^3\\106625 &= 5^3 \cdot 853 \text{ divisible by }125 =5^3 \end{align}$$What's interesting about sequences like this is that the numbers derive their membership via the groups to which they belong. For convenience, as in the case of OEIS A178919, only the first number in the group is listed. It is the relationship between the numbers in the group that are important. In the case of OEIS A178919 the numbers form a group of three that are consecutive and divisible by consecutive squares. Thus we have in the case of 27423:$$ \text{consecutive integers -->}\\ \frac{27423}{9} \, \frac{27424}{16} \, \frac{27425}{25} \\ \text{consecutive squares -->} $$or in the case of 106623:$$ \text{consecutive integers -->}\\ \frac{106623}{27} \, \frac{106624}{64} \, \frac{106625}{125} \\ \text{consecutive cubes -->} $$It would be interesting to explore divisibility using criteria other than divisibility by consecutive squares or cubes. What about divisibility of three consecutive integers by three consecutive fibonacci numbers (0, 1, 1, 2, 3, 5, 8, ...)? Well, if we ignore 0 and 1 and start with 2, it turns out that a great many groups of three qualify, most of which are divisible by 2, 3 and 5. The first of these begins with 8:$$ \begin{align} 8 &= 2^3 \text{ divisible by fibonacci number }2\\9 &= 3^2 \text{ divisible by fibonacci number } 3\\10 &= 2 \cdot 5 \text{ divisible by fibonacci number } 5 \end{align}$$There are 4417 such groups of three in the range up to 100,000, so they are very common. If we exclude 2, 3 and 5 and begin instead with 8, then the groupings of three become far less common (only 60 in the range up 100,000). The first of these begins with 376 (permalink):$$ \begin{align} 376 &= 2^3 \cdot 47 \text{ divisible by fibonacci number } 8\\377 &= 13 \cdot 29 \text{ divisible by fibonacci number }13\\378 &= 2 \cdot 3^3 \cdot 7 \text{ divisible by fibonacci number } 21 \end{align}$$This is clearly a topic worthy of further research.

Amicable Tuples

When I think of amicable numbers, a pair of numbers come to mind: 220 and 284. They have the property that the sum of the proper divisors of 220 equals 284 and vice versa. This is an example of an amicable 2-tuple. The relationship between 220 and 284 can be expressed as:$$ \sigma_1(220)=\sigma_1(284)=220+284$$This means that any amicable 2-tuple, let's say \( (x, y) \), has the property that:$$ \sigma_1(x)=\sigma_1(y)=x+y $$The next amicable pair or 2-tuple is (1184, 1210). A list of numbers that form amicable pairs is given by OEIS A063990:

A063990

Amicable numbers.

The initial members are:

220, 284, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368, 10744, 10856, 12285, 14595, 17296, 18416, 63020, 66928, 66992, 67095, 69615, 71145, 76084, 79750, 87633, 88730, 100485, 122265, 122368, 123152, 124155, 139815, 141664, 142310

The sequence lists the amicable numbers in increasing order. Note that the pairs \( (x, y) \) are not necessarily adjacent to each other in the list. The first time a pair ordered by its first element is not adjacent is \(x\) = 63020, \(y\) = 76084 which correspond to a(17) and a(23), respectively.

This leads us on to amicable 3-tuples, \( (x, y,z) \), that have the property:$$ \sigma_1(x)= \sigma_1(y)= \sigma_1(z)=x+y+z$$The first amicable triple or 3-tuple is (1980, 2016, 2556). My diurnal age today, 27312, is part of the amicable 3-tuple (27312, 21168, 22200).  In general, we can call a finite set \( (x_1, x_2, \dots, x_k) \) of natural numbers (the \(x_i\) are pairwise distinct), an amicable \(k\)-tuple iff$$  \sigma_1(x_1)= \sigma_1(x_2)=\dots =\sigma_1(x_k)=x_1+x_2+...+x_k $$For \(k\)=1, the only possible amicable one-tuple is (1).

OEIS A255215 lists numbers that belong to at least one amicable tuple and the initial members are:

1, 220, 284, 1184, 1210, 1980, 2016, 2556, 2620, 2924, 5020, 5564, 6232, 6368, 9180, 9504, 10744, 10856, 11556, 12285, 14595, 17296, 18416, 21168, 22200, 23940, 27312, 31284, 32136, 37380, 38940, 39480, 40068, 40608, 41412, 41952, 42168, 43890, 46368, 47124

The Esucarys Mapping Revisited

It was on the 15th February 2021 that I made my first post about the Esucarys Mapping which is related the Collatz or 3\(x\)+1 mapping but with an extra twist. Let's revisit what I wrote back then.

The Esucarys sequence derives its name from a reversal of "Syracuse", with the generating rule being that for the Syracuse (3\(x\)+1 or Collatz) sequence followed by a reversal. 247 is the only known fixed point of the Esucarys sequence. Very few numbers map to 247.

The members of this sequence, up to 40000, are:

247, 1247, 1484, 2473, 4859, 5087, 5738, 7318, 7484, 9563, 9682, 9694, 9938, 11247, 12189, 12473, 14840, 14842, 15209, 15610, 16274, 16563, 16750, 16798, 17609, 19168, 20019, 21885, 24733, 26251, 27123, 27125, 29156, 30076, 30524, 32614

Back when I made that post my diurnal age was 26251 and it was only today that my diurnal age reached the next term, 27125, in this sequence (OEIS A129133). This latter number requires only five steps to reach 247. The steps are:

27123, 7318, 9563, 9682, 1484, 247

The trajectory is shown in Figure 1.

Figure 1

The progression reached thus:

• 27123 --> 81370 (multiply by 3 & add 1 since number is odd)
• 81370 --> 7318 (reverse number)
• 7318 --> 3659 (divide by two since number is even)
• 3659 --> 9563 (reverse number)
• 9563 --> 28690 (multiply by 3 and add 1 since number is odd)
• 28690 --> 9682 (reverse number)
• 9682 --> 4841 (divide by 2 since number is even)
• 4841 --> 1484 (reverse number)
• 1484 --> 742 (divide by 2 since number is even)
• 742 --> 247 (reverse number)

Since 247 --> 742 --> 247 we are stuck. Note that certain numbers will produce infinite loops but they don't centre on a fixed point. For example consider the numbers 3 and 13:

3 --> 10 --> 1 --> 4 --> 2 --> 1 --> 4

13 --> 40 --> 4 --> 2 --> 1 --> 4

What's interesting about 27123 is that the next odd number, 27125, is also a member of the sequence. The progression for this number also involves five steps and is:

27125, 67318, 95633, 9682, 1484, 247

The maximum values reached by the trajectory of both these numbers is tiny compared to the previous singleton (26251, see earlier post). Pairings like 27123 and 27125 are relatively rare. Up to one hundred thousand, the only ones are:

• 14840 and 14842
• 27123 and 27125
• 74840 and 74842
• 82823 and 82825

 Figure 2 shows a plot of the sequence members up to one hundred thousand.

Figure 2

Prime Emirp Pair Averages

An emirp is a prime that remains prime when its digits are reversed. The prime and its reversal must be different and so this excludes palindromic primes like 101. The smallest emirp is 13 that, when reversed, gives 31 which is also prime.  By "Prime Emirp Pair Averages", I mean primes that are the average of an emirp pair. The first such prime is 11311, a palindromic prime, and it is the average of the emirp pair 10321 and 12301. Thus$$11311=\frac{10321+12301}{2}$$These sorts of primes form OEIS A178581:

A178581

Primes that are the average of the members of emirp pairs.

The initial members are:

11311, 12721, 13831, 14741, 16061, 16561, 17471, 18481, 20507, 21107, 21407, 21617, 21817, 22727, 23027, 23227, 23327, 23537, 24137, 24547, 24847, 25147, 25247, 25447, 25657, 26357, 27067, 27367, 28277, 34543, 34843, 35153, 35353

Some of these primes are the average of more than one emirp pair. The first such prime is 14741, another palindromic prime, and it is the average of two emirp pairs: the first pair being 10781 and 18701 and the second being 13751 and 15731. $$ \begin{align} 14741&=\frac{10781+18701}{2}\\ &=\frac{13751+15731}{2} \end{align}$$The first such prime that is the average of three emirp pairs is 24547. It is the average of (11083, 38011), (12073, 37021) and (18013, 31081). Thus$$ \begin{align} 24547&=\frac{11083+38011}{2}\\ &=\frac{12073+37021}{2} \\ &= \frac{18013+31081}{2} \end{align}$$The first such prime that is the average of four emirp pairs is 25447. It is average of (10993, 39901), (13963, 36931), (17923, 32971) and (18913, 31981). Thus $$ \begin{align} 25447 &=\frac{10993+39901}{2} \\ &= \frac{13963+36931}{2}\\ &= \frac{17923+32971}{2} \\ &= \frac{18913+31981}{2} \end{align}$$These primes form OEIS A178587 (permalink):

A178587

Primes that are the average of the members of more than one emirp pair.

The initial members are:

14741, 22727, 23327, 24547, 25447, 27067, 28277, 42929, 63541, 65761, 85453, 1217171, 1221221, 1227271, 1243421, 1245421, 1246471, 1250521, 1253521, 1257521, 1261571, 1271671, 1283771, 1327231, 1335331, 1338331, 1339381 

What's noteworthy with this sequence is the gap between 85453 and the next prime, 1217171. That's quite a gap. I was alerted to these sorts of primes because the number associated with my diurnal age, 27067, is a member of OEIS A178587, as well as OEIS A178581 of course. For want of a better name, a prime of this sort might be called a PEPA prime with the acronym standing for Prime Emirp Pair Average.