A DeepSeek Puzzle Sequence

3, 5, 4, 4, 6, ?

Following my somewhat disappointing experience with Google's Gemini in generating puzzle sequences (see Gemini Sequences), I thought I'd test out DeepSeek. To be fair I gave the AI rather more detail as regards what I wanted here is the interchange (my questions are in blue, DeepSeek in red).

Can you create a sequence of five numbers according to an undisclosed pattern and I have to guess the sixth number. The pattern may be based on numbers or the names of the numbers or even how the numbers are spoken. Whatever the underlying pattern, it should be challenging to uncover, requiring resourcefulness, creativity and imagination on the part of the person attempting to find the sixth number.

Sure! Here's a sequence of five numbers following an undisclosed pattern. Your task is to guess the sixth number:

**Sequence:** 3, 5, 4, 4, 6, ?

Good luck! Let me know your guess, and I’ll tell you if you’re correct or provide hints if needed.

Add 2, subtract 1, subtract 0. If this pattern is followed then 4+3=6 (the fifth number) and 6-1=5 so 5 is the sixth number.

Interesting attempt! However, the pattern you’ve proposed doesn’t align with the one I had in mind. Let me give you a hint to help you uncover the true pattern:

**Hint:** The sequence is based on the *number of letters* in the English words for the numbers. For example:

- "Three" has 5 letters.

- "Five" has 4 letters.

- "Four" has 4 letters.

- "Six" has 3 letters.

Now, using this pattern, can you guess the sixth number? Let me know your answer! 😊

3 has 5 letters, 4 has 4 letters, so 6 has 3 letters. So my answer is 3. 

Great effort! However, let’s revisit the sequence and the hint to clarify the pattern:

**Sequence:** 3, 5, 4, 4, 6, ?

**Hint:** The sequence is based on the *number of letters* in the English words for the numbers:

- "Three" has 5 letters → 5

- "Five" has 4 letters → 4

- "Four" has 4 letters → 4

- "Six" has 3 letters → 3

So, following this pattern, the next number should correspond to the number of letters in the word for the previous number. Since "six" has 3 letters, the next number should be **3**.

You were correct! The sixth number is indeed **3**. Well done! 🎉

So this wasn't a bad puzzle and as can be seen I struggled initially. I'll be sure to try DeepSeek again in the near future.

Primorials and the Sigma Function

I noticed that the sum of divisors (64680) of the number (27708) that represents my diurnal age today has the following factorisation:$$64680 = 2^3 \times 3 \times 5 \times 7^2 \times 11$$These prime factors, ignoring multiplicity, represent the factorisation of a primorial, in this case the primorial 2310:$$2310 = 2 \times 3 \times 5 \times 7 \times 11$$This got me wondering what other numbers in the range up to 40000 have a sum of divisors whose prime factors, again without multiplicity, form the primorial 2310. It turns out that there are 267 such numbers. They are (permalink):

1538, 2180, 2309, 2456, 2636, 2834, 3488, 3688, 3845, 4469, 4472, 4614, 4618, 4796, 4988, 5276, 6152, 6158, 6540, 6927, 7085, 7368, 7412, 7690, 7908, 7916, 8424, 8459, 8502, 8567, 8759, 8780, 8903, 8938, 9047, 9236, 9239, 9396, 9848, 9956, 10028, 10148, 10464, 10766, 11064, 11336, 11414, 11535, 11545, 11549, 11666, 11876, 11954, 12280, 12447, 12644, 13073, 13180, 13196, 13369, 13407, 13416, 13544, 13624, 13854, 13859, 14048, 14104, 14148, 14170, 14388, 14776, 14964, 14972, 15196, 15260, 15308, 15380, 15395, 15398, 15587, 15828, 16163, 16211, 16340, 16578, 16918, 17134, 17147, 17192, 17440, 17518, 17687, 17806, 17876, 17999, 18017, 18094, 18113, 18203, 18209, 18440, 18452, 18456, 18472, 18474, 18478, 19116, 19316, 19838, 19988, 19994, 20488, 20492, 20789, 21088, 21116, 21146, 21242, 21255, 21276, 22236, 22301, 22345, 22360, 22518, 22852, 23070, 23090, 23098, 23099, 23108, 23220, 23748, 23756, 23980, 24089, 24416, 24608, 24632, 24894, 24940, 25064, 25377, 25399, 25409, 25628, 25701, 25724, 25727, 25816, 26144, 26146, 26277, 26340, 26380, 26396, 26568, 26709, 26738, 26814, 26915, 27016, 27019, 27141, 27199, 27323, 27352, 27359, 27383, 27708, 27717, 27718, 27956, 28340, 28404, 28535, 28552, 28836, 28996, 29165, 29222, 29544, 29852, 29868, 29885, 30014, 30017, 30084, 30186, 30302, 30444, 30508, 30537, 30760, 30780, 30788, 30790, 30916, 30956, 31174, 31283, 31304, 31529, 31676, 31928, 31937, 32298, 32326, 32357, 32422, 32591, 32981, 33256, 33368, 33497, 33572, 33836, 33869, 33939, 34008, 34242, 34268, 34294, 34635, 34647, 34649, 34916, 34998, 35036, 35093, 35180, 35374, 35612, 35628, 35699, 35752, 35862, 35948, 35998, 36034, 36077, 36143, 36188, 36226, 36406, 36418, 36419, 36437, 36557, 36840, 36932, 36956, 37060, 37152, 37932, 38368, 38495, 38498, 38597, 38804, 39219, 39253, 39540, 39580, 39588, 39904

What caught my attention in this sequence of numbers was a pair of numbers that will be coming up for me in a little over a week. The numbers are 27717 and 27718. This got me to wondering if there were other such number pairs in the range up to 40000 and it turns out that there are. The other two are (23098, 23099) and (36418, 36419). See Table 1 for the details. 

Table 1

27718 also has the interesting property that its cototient has the same prime factors as 2310 since the totient is 13858 and thus the cototient is$$27718 - 13858 = 13860 = 2^2 \times 3^2 \times 5 \times 7 \times 11$$In the range up to 40000, the only numbers with this property are 4618, 9236, 18472, 18478, 23098, 27718 and 36956 (permalink). 

Anyway getting back on track, these pairs got me thinking about runs of three consecutive numbers and perhaps higher runs. I extended the range up to one million and in that range there are seven triplets of numbers whose sum of divisors consists form the factors of the primorial 2310. See Table 2.

Table 2

Looking at Table 2 it can be seen that there are two groups of quadruplets. See Table 3.

Table 3

Let's just double check the last quadrupets 692994, 692995, 692996 and 692997. See Table 4.

Table 4

So in terms of my diurnal age what's of interest is that the number pair 27717 and 27718 is coming up in a little over a week and its members share the interesting property discussed in this post. Let's move on to the primorial 210 = 2 x 3 x 5 x 7. In the range up to 40000, there are 1945 numbers with sums of divisors with prime factors (considered without multiplicity) that multiply together to give the primorial 210. I won't list them all but here is a permalink.

Restricting ourselves to the range up to 40000, we do get two groups of quintuplets. They are 20154 to 20158 and 29395 to 29399. 

Quadruplets are more numerous of course and Table 5 shows these.

Table 5: permalink

Groups of triplets are shown in Table 6.

Table 6: permalink

The pairs are too numerous to list here but this is a permalink.

Gemini Sequences

I asked Gemini to do the following (my prompts in blue, Gemini's responses in red):

Can you generate the first five terms of a sequence according to a clever algorithm that I have to discover in order to find the sixth term. Don’t tell me what it is but let me supply an answer which you can decide is right or wrong.

Here's what it came up with:

Okay, here are the first five terms of the sequence:

2, 6, 15, 30, 56

What is the sixth term?

This is not a bad effort. The next term becomes apparent when we consider the factorisations of the numbers in the sequence:

•   2 = 1 x 2
•   6 = 2 x 3
• 15 = 3 x 5
• 30 = 5 x 6
• 56 = 7 x 8

The rule clearly is non-composite number multiplied by the next consecutive number. Here, after 7, the next non-composite number is 11 followed by 12 and so 132 = 11 x 12 is the sixth member of the sequence. The use the term non-composite rather than prime because 1 is not prime.

I was thinking of sequences today since I spotted this way in an early morning tweet. Find the next number in this sequence and 42 is not the correct answer:

12, 15, 21, 24, 30, 33, 39, ...

Looking at the differences between successive terms, we see a clearly alternating pattern of adding 3 and then adding 6 to generate successive terms. Following this pattern, then 42 would indeed be the next term. This would be quite valid except we are told that this is not an answer that is acceptable.

Looking more closely at the sum of the numbers' digits we see that the algorithm required to generate the next number from the previous is number + sum of digits. Thus after 39 we must have 39 + 12 = 52.

I prompted Gemini again:

Could you provide another sequence of five numbers for which I have to guess the sixth number?

Here's another sequence for you:

1, 3, 7, 15, 31

What is the next term? 

Hmmm. That was a little too easy. The next term is 63 according to the rule \(2^n-1\) for \(n=1,2,3,4,5,6, \dots\). I complained:

Except that last one was a little too easy. Can you provide another sequence that is more challenging?

You got it. Here's a more challenging sequence:

3, 10, 21, 36, 55

What is the next term?

Well, again not too hard when we factorise the numbers.

•   3 = 1 x 3
• 10 = 2 x 5
• 21 = 3 x 7
• 36 = 4 x 9
• 55 = 5 x 11

Here the rule is \(T_n=n \times (n+2) \) and so the next term is 78 = 6 x 13. Except it isn't. At least according to Gemini. Here's were the AI had a serious glitch as the following exchange illustrates:

Incorrect.  Try again!  The next term isn't 78.

OK I give up but I would argue that 78 is a valid number since we have 3 = 1 x 3, 10 = 2 x 5, 21 = 3 x 7, 36 = 4 x 9 and 55 = 5 x 11. Logically, the next number should be 6 x 13 = 78 since according to the rule T_n=n x (2n+1). What do you consider the correct response and how did you arrive at your answer?

You're absolutely right that 78 fits a pattern!  Your logic is sound, and you've identified a valid sequence. My apologies – I was thinking of a different pattern when I generated the sequence, and I incorrectly marked your answer as wrong.  78 is a perfectly acceptable and logical next term. 

The pattern I had in mind was based on the sums of consecutive odd numbers, specifically, the nth term is equal to n(2n+1).
3 = 1 * (2*1 + 1)
10 = 2 * (2*2 + 1)
21 = 3 * (2*3 + 1)
36 = 4 * (2*4 + 1)
55 = 5 * (2*5 + 1)
So the next term would be 6 * (2*6 + 1) = 6 * 13 = 78.

So, you were correct and you've actually identified the pattern I had in mind! My apologies again for the confusion.  It seems I wasn't as clever as I thought I was! 😊

Oh dear, Gemini at least will not be taking over the world anytime soon with that sort of thinking. Anyway, it was an interesting little experiment but disturbing in that I provided the correct answer (that's all that was required) and yet it said that was wrong. Here is what Gemini said when I supplied 78 as the answer:

Additionally the problems that Gemini came up with were not all that imaginative. Perhaps if I'd fed it some challenging sequences as examples it might have done better.

Record Prime v. Non-Prime Digit Trajectory Lengths

In a recent post titled Prime and Non-Prime Digit Sequence in November of 2024, I explained that:

Instead of dealing with odd and even digits, trajectories involving prime and non-prime digit sums can be considered. The prime digits are 2, 3, 5 and 7 while the non-prime digits are 0, 1, 4, 6, 7 and 9.

For example, the number associated with my diurnal age today is 27632 and it has the following trajectory of 3 steps under the prime and non-prime algorithm (permalink):

27632 --> 27640 --> 27639 --> 27636 --> 27636

The trajectory terminates when the number has a sum of primes and sum of non-primes that are equal, here 2 + 7 + 3 = 6 + 6. 

🅃🅁🄰🄹🄴🄲🅃🄾🅁🅈 

What I want to look at this post are the trajectory lengths of numbers under this prime and non-prime digit algorithm. Table 1 shows a plot of the results up to 100,000.

Figure 1: permalink

The lines are not vertical of course. It just means that the new records are clumped together. Table 1 shows the clumping as we jump from 8185 to 14067 which is where some brief clumping occurs (14067, 14077 and 14085) before we jump to 19165 which is where the real clumping occurs, up to 19452. After this there is massive jump 49370 where more clumping occurs.

Table 1: permalink

From the perspective of my diurnal age analysis, the last record occurred when I was 19452 days old and here was its trajectory:

19452, 19445, 19432, 19423, 19414, 19395, 19384, 19365, 19357, 19362, 19351, 19348, 19329, 19315, 19312, 19306, 19293, 19279, 19269, 19246, 19228, 19214, 19201, 19192, 19174, 19166, 19143, 19131, 19122, 19115, 19108, 19089, 19062, 19048, 19026, 19012, 19003, 18996, 18963, 18942, 18922, 18908, 18882, 18859, 18838, 18816, 18792, 18783, 18776, 18775, 18785, 18780, 18770, 18775

The number 19452 is captured by the vortex 18775, 18785, 18780, 18770, 18775. So no more records for me as I am not going to live to see a diurnal age of 49370.

Super Attractors

In my own private terminology, I deem a number an odd-even attractor if its sums of odd digits and even digits are the same. I use the term attractor because numbers that are not attractors are "attracted" to such numbers. For example, let's take the case of 134. Here the sum of the odd numbers is 1 + 3 = 4 and the sum of the even numbers is 4. Thus it is an odd-even attractor. 

Let's take a number like 122 that is not an odd-even attractor. The sum of the even digits (4) exceeds the sum of the odd digits (1).  The difference between odd and even digits is 1 - 4 = -3 and this will be added to the original number to get 119. Now the sum of the odd numbers (11) exceeds that of the non-existent even numbers (0) and this is added to 119 to get 130. Repeating the process we get 134 which is an attractor.

In this system, I've chosen to subtract the sum of the even digits from the sum of the odd digits. This is quite arbitrary and I could have chosen to subtract the sum of the odd digits from the even digits but for odd-even or even-odd attractors this doesn't matter. A similar system can be adopted for prime and non-prime digits. The prime digits are 2, 3, 5 and 7 whereas the non-prime digits are 0, 1, 4, 6, 8 and 9. A number wherein the sum of the prime digits equals that of the non-prime digits is called, in my nomenclature, a prime-non-prime attractor. An example would be 358 where 3 + 5 = 8.

A number that is not a prime-non-prime attractor is 356. Here the sum of prime digits is 8 and the sum of the non-prime digits is 6. We chose to subtract the sum of non-prime digits from the sum of the prime digits to get 2 which we add to 356 to get 358 which is a prime-non-prime attractor.

The question that I was interested in is how many numbers are both odd-even attractors and prime-non-prime attractors? We might term these super attractors. In the range up to 40000, there are 222 such numbers (permalink) and they are:

112, 121, 211, 336, 358, 363, 385, 538, 583, 633, 835, 853, 1012, 1021, 1102, 1120, 1201, 1210, 2011, 2101, 2110, 3036, 3058, 3063, 3085, 3306, 3360, 3445, 3454, 3467, 3476, 3508, 3544, 3580, 3603, 3630, 3647, 3674, 3746, 3764, 3805, 3850, 4345, 4354, 4367, 4376, 4435, 4453, 4534, 4543, 4556, 4565, 4578, 4587, 4637, 4655, 4673, 4736, 4758, 4763, 4785, 4857, 4875, 5038, 5083, 5308, 5344, 5380, 5434, 5443, 5456, 5465, 5478, 5487, 5546, 5564, 5645, 5654, 5667, 5676, 5748, 5766, 5784, 5803, 5830, 5847, 5874, 6033, 6303, 6330, 6347, 6374, 6437, 6455, 6473, 6545, 6554, 6567, 6576, 6657, 6675, 6734, 6743, 6756, 6765, 6778, 6787, 6877, 7346, 7364, 7436, 7458, 7463, 7485, 7548, 7566, 7584, 7634, 7643, 7656, 7665, 7678, 7687, 7768, 7786, 7845, 7854, 7867, 7876, 8035, 8053, 8305, 8350, 8457, 8475, 8503, 8530, 8547, 8574, 8677, 8745, 8754, 8767, 8776, 10012, 10021, 10102, 10120, 10201, 10210, 11002, 11020, 11200, 12001, 12010, 12100, 20011, 20101, 20110, 21001, 21010, 21100, 30036, 30058, 30063, 30085, 30306, 30360, 30445, 30454, 30467, 30476, 30508, 30544, 30580, 30603, 30630, 30647, 30674, 30746, 30764, 30805, 30850, 33006, 33060, 33600, 34045, 34054, 34067, 34076, 34405, 34450, 34504, 34540, 34607, 34670, 34706, 34760, 35008, 35044, 35080, 35404, 35440, 35800, 36003, 36030, 36047, 36074, 36300, 36407, 36470, 36704, 36740, 37046, 37064, 37406, 37460, 37604, 37640, 38005, 38050, 38500

Figure 1 shows the rather uneven distribution of such numbers in the range up to 40000:

Figure 1

All of the numbers greater than 10000 contain the digit 0. Attractors are very much base-specific and thus fall into the realm of recreational mathematics. The big gaps occur between 12100 and 20011 and 21100 and 30036. Numbers that are not attractors of either sort but are close to super attractors do not necessarily end up attracted to the nearest attractor. 

Take 30035 that is next to the super attractor 30036. Here is its prime-non-prime trajectory:

\(30035 \rightarrow 30046 \rightarrow 30039 \rightarrow 30036 \rightarrow 30036\)

While it ends up at the nearby super attractor, the same is not true when subjected to the odd-even trajectory:

\(30035 \rightarrow 30046 \rightarrow 30039 \rightarrow 30054 \rightarrow 30058 \rightarrow 30058\)

It ends up at the more distant super attractor 30058.

Building Block Numbers

The prime numbers are considered the building blocks of the positive integers because every composite number can be expressed as a product of prime numbers. However, the OEIS A086424 offers an alternative set of building blocks:

A086424 Numbers needed to generate all other natural numbers, only allowing multiplication and addition. Each number can be used only once.

These numbers are: 

1, 2, 4, 11, 25, 64, 171, 569, 3406, 27697, 243374, 1759619, 28381401, 222323189, 3416307938, 26838745347

Here is an extract from the OEIS comments:

• 10 is not in the sequence because (4+1)*2 = 10.
• 11 is in the sequence because there is no way to get 11 by using the earlier terms.
• 509 is not in the sequence because
  509 = (1+25)*(2+11)+171.

The number associated with my diurnal age today is 27697 and this is the first number that cannot be expressed in terms of the numbers 1, 2, 4, 11, 25, 64, 171, 569 and 3406 using only multiplication or addition and using each number only once. From now on, up to but not including 243374, every number can be expressed in terms of 1, 2, 4, 11, 25, 64, 171, 569, 3406 and 27697. This is a big deal.

GEMINI WAS ABLE TO CHANGE PARI CODE
TO PYTHON AFTER A FEW TRIES

The OEIS comments offer some PARI code to generate this list of numbers and, after many errors on the part of Gemini, I was able to get to produce some working PYTHON code that will do the same job, albeit slowly. Here is the code:

from sage.all import *

def Ww(v):

    """

    Calculates the Ww function for a given list of integers.

    Args:

        v: A list of integers.

    Returns:

        A list of integers resulting from the Ww function.

    """

    if len(v) == 2:

        return [v[0], v[1], v[0] + v[1], v[0] * v[1]]

    else:

        V = []

        for i in range(len(v)):

            for j in range(i + 1, len(v)):

                t = v[:i] + v[i+1:j] + v[j+1:]

                if t:

                    V.extend(Ww(t + [v[i] + v[j]]))

                    V.extend(Ww(t + [v[i] * v[j]]))

        return sorted(set(V))  # Remove duplicates

a = [Integer(1), Integer(2), Integer(4)]

for n in range(3, 10):

    V = Ww(a)

    for i in range(2 * a[-1], len(V) + 1):

        if V[i - 1] > i:

            a.append(i)

            print(f"a = {a}")

            break

    else:

        print(f"No solution found for n = {n}")  

 

EVERY NUMBER UP TO BUT NOT INCLUDING 27697

CAN BE EXPRESSED IN TERMS OF

1, 2, 4, 11, 25, 64, 171, 569 and 3406

USING ONLY ADDITION AND MULTIPLICATION

AND USING ONE NUMBER ONLY ONCE

Of course the execution of this code on SageMathCell will quickly time out and it will only generate the numbers up to 171. However, running it in a Jupyter notebook on an M1 Macbook Air, it will generate the numbers up to 3406 fairly quickly but after an hour or so it never got to 27697 so I stopped it. 

I asked Gemini to come up with an algorithm in Python to express a given number in terms of these new "building blocks" but it consistently failed so I gave up. However, it will be easy initially for numbers greater than 27697. For example:

• 27698 = 27697 + 1
• 27699 = 27697 + 2
• 27700 = 27697 + 2 + 1
• 27701 = 27697 + 4 etc.

I'll continue to do this as an exercise associated with my daily number analysis. Figure 1 shows an analysis of the building block numbers and their factors. It can be seen that the only prime numbers are 2, 11, 569 and 27697. It should be noted that the property these numbers have collectively is NOT base dependent.

Figure 1

These particular building blocks are remarkably economical considering how many unique primes we would require to "build" all the numbers from 2 to 27697. It is in fact 3020. These blocks only require nine blocks: 1, 2, 4, 11, 25, 64, 171, 569 and 3406. They become even more economical for larger numbers. For example in the range up to 243373 (one less than 243374), we require 21494 unique primes but only ten of our new building blocks: 1, 2, 4, 11, 25, 64, 171, 569, 3406 and 27697. However, unlike the prime building blocks, representations using these new building blocks are not unique. For example, 12 can be represented as 11 + 1 or (1 + 2) * 4. Remember, bracketing is allowed.

Odd and Even Digit Averages

What do the following 98 numbers have in common (permalink)?

102, 123, 147, 234, 258, 306, 345, 369, 456, 567, 678, 789, 1012, 1034, 1056, 1078, 1223, 1245, 1267, 1289, 1447, 1469, 2334, 2356, 2378, 2558, 3036, 3058, 3445, 3467, 3489, 3669, 4556, 4578, 5667, 5689, 6778, 7889, 10004, 10022, 10036, 10112, 10167, 10234, 10356, 10478, 10667, 11125, 11233, 11247, 11459, 11477, 12223, 12278, 12345, 12467, 12589, 12778, 13349, 13457, 14447, 14555, 14569, 14677, 15699, 16779, 20238, 20346, 22236, 22344, 22358, 22588, 23334, 23389, 23456, 23578, 23889, 24568, 25558, 25666, 25788, 30048, 30066, 30336, 30444, 30458, 30566, 33347, 33455, 33469, 33699, 34445, 34567, 34689, 35679, 36669, 36777, 36899

First and foremost, all the numbers have their digits arranged in ascending order. Apart from that however, what else do they have in common? Let's look at the largest member of the sequence: 36899. It has odd digits of 3, 9, 9 and even digits of 6, 8. What are the averages of the odd and even digits? Well, (3 + 9 + 9)/3 = 7 and (6 + 8)/3 = 7 and so the average of the odd and even digits is the same. This is true of all the numbers. The reason that the digits have been arranged in ascending order is that these might be called "root" numbers because any number that is a permutation of the digits of the one of these numbers will have the property that its average of odd and even digits is the same.

In the range up to 40000, there are 1886 numbers that satisfy this equal average criterion. This comprises 4.71% of the range. Here is a permalink that will display these numbers. If we want to apply additional criteria, then this total of 1886 can be reduced significantly. For example, let’s require that the numbers be prime as well as the average of odd and even digits. The first such number would be 1223 which is prime and where the odd average is (1 + 3)/2 = 2 and the even average is (2 + 2)/2 = 2. There are 97 such numbers in the range up to 40000 (permalink):

1223, 1289, 2213, 2819, 2837, 3041, 3221, 3467, 4013, 4637, 4673, 4691, 5689, 5869, 6473, 6491, 7283, 7643, 7687, 7823, 7867, 8219, 8237, 8273, 8291, 8677, 9281, 9461, 10243, 11251, 12043, 12511, 12589, 14767, 15121, 15289, 15649, 16477, 16747, 17467, 17827, 20143, 20341, 20431, 21313, 21589, 21787, 21859, 22123, 23041, 23131, 23311, 23857, 23893, 24103, 25111, 25189, 25819, 25873, 25981, 27583, 27817, 28393, 28537, 28573, 28591, 28753, 28771, 28933, 29383, 29581, 29833, 29851, 30241, 31123, 31231, 31321, 32401, 32587, 32839, 32983, 33211, 33289, 33469, 33829, 34369, 34693, 34963, 36457, 36493, 36899, 36943, 38239, 38329, 38699, 38923, 39869

Neither of these two sequences is registered in the OEIS and I certainly won't be proposing them to the cretins who oversee it. However, the sequences are in my own private database. Here is the link. The averages themselves do not need to be integers. For example 3863503 has an odd average of (3 + 3 + 5 + 3)/3 = 14/3 and an even average of (8 + 6 + 0)/3 = 14/3. However, for all numbers in the range up to 40000, the averages are all integers.

Of course, we need not consider odd and even digits. We could consider the average of prime (2, 3, 5 and 7) and non-prime (0, 1, 4, 6, 8, 9) digits. An example of a number in which the average of the prime and non-prime digits is the same would be 39760 where (3 + 7)/2 = 5 and (9 + 6 + 0)/3 = 5. Here is a permalink to generate these numbers.