Primeval Numbers

Today I learned what a $\textbf{primeval number}$ is and it's not surprising that I haven't heard of this type of number before. They are after all rather light on the ground. Here's a definition:

Primeval number: a prime which "contains" more primes in it than any preceding number. Here "contains" means may be constructed from a subset of its digits.

Table 1 shows all the primeval numbers less than 100,000.

Table 1

These numbers form OEIS A072857. After 13679 there is huge jump to 100279.

1, 2, 13, 37, 107, 113, 137, 1013, 1037, 1079, 1237, 1367, 1379, 10079, 10123, 10136, 10139, 10237, 10279, 10367, 10379, 12379, 13679, 100279, 100379, 101237, 102347, 102379, 103679, 123479, 1001237, 1002347, 1002379, 1003679, 1012349, 1012379, 1023457, 1023467, 1023479, 1234579, 1234679, 10012349

It can be noted that all these numbers are plaindromes or numbers whose digits are in increasing order as required by the definition. For example, 31 contains three primes (3, 13 and 31) as does 13 but the former is not listed because 13 is the first prime to contain three primes.

I was made aware of these primeval numbers via a property of the number associated with my diurnal age today, 27724, that earns it admission into OEIS A173052:

A173052  partial sums of A072857 (primeval numbers: numbers that set a record for the number of distinct primes that can be obtained by permuting some subset of their digits).

These partial sums are:

1, 3, 16, 53, 160, 273, 410, 1423, 2460, 3539, 4776, 6143, 7522, 17601, 27724, 37860, 47999, 58236, 68515, 78882, 89261, 101640, 115319, 215598, 315977, 417214, 519561, 621940, 725619, 849098, 1850335, 2852682, 3855061, 4858740, 5871089

Celebrating 27720

It's not often that numbers as large as 27720 attract 632 entries in the Online Encycopedia of Integer Sequences (OEIS). By contrast, 27719 attracts 31 entries and 27721 attracts 27 entries. So what's so special about 27720?

Well, it has lots of interesting properties. Let's look at some of them. 

$\textbf{PROPERTY 1}$

The very first entry in the database is OEIS A002182:

A002182  Highly composite numbers: numbers $n$ where d($n$), the number of divisors of $n$  increases to a record.

The initial record holders, up to 40000, are as follows where we see 27720 is a member:

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720

Table 1 shows the details:

Table 1: permalink

$\textbf{PROPERTY 2}$

In a similar vein is OEIS A004394 where 27720 also features:

A004394    superabundant numbers: $n$ such that $\sigma(n)/n > \sigma(m)/m$ for all $m < n$, $\sigma(n)$ being A000203(n), the sum of the divisors of $n$.

The initial members are as follows with most being the same as for OEIS A002182:

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 10080, 15120, 25200, 27720

Table 2 shows the details:

Table 2: permalink

$\textbf{PROPERTY 3}$

Another interesting property of the number arises from its appearance in the denominator of the progressive sum of the harmonic numbers. These denominators constitute OEIS A002805.

A002805    denominators of harmonic numbers $\text{H}(n) =\displaystyle \sum_{i=1} ^n \dfrac{1}{i}$

The first terms in the sequence are 1, 2, 6, 12, 60, 20, 140, 280, 2520, 2520, 27720, 27720. Table 3 shows the details.

Table 3: permalink

$\textbf{PROPERTY 4}$

The number also arises from a quite simple recurrence relation:

A052542     $\text{a}(n) = 2 \times \text{a}(n-1) + \text{a}(n-2), \text{ with } \text{a}(0) = 1, \text{a}(1) = 2, \text{a}(2) = 4$

The initial members of the sequence are 1, 2, 4, 10, 24, 58, 140, 338, 816, 1970, 4756, 11482, 27720 (permalink).

$\textbf{PROPERTY 5}$

Since $27720 = 2^3 \times 3^2 \times 5 \times 7 \times 11$, it is 12 times the product of the primorial number $2310 = 2 \times 3 \times 5 \times 7 \times 11$ and this qualifies it for membership in OEIS A129912 because 12 is itself a product of primorials viz. 2 x 6.

A129912 numbers that are products of distinct primorial numbers (see A002110).

The initial members of the sequence (with the primorials themselves included) can be generated using this permalink:

1, 2, 6, 12, 30, 60, 180, 210, 360, 420, 1260, 2310, 2520, 4620, 6300, 12600, 13860, 27720, 30030, 37800

Initially I misinterpreted this sequence as meaning numbers that are multiples of primorials but this is not the case. Instead the multiples themselves must be products of primorials and this is far more restrictive.

Primorial Number Base Revisited

On the 14th February 2021, now over four years ago, I created a post on this blog about the Primorial Number System. Since then, I've thought very little about it but today's number (associated with my diurnal age) reminded me once again of this number system. The number is $\textbf{27718}$ and it is a member of OEIS A333703:

A333703   Numbers $k$such that $k$ divides the sum of digits in primorial base of all numbers from $1$ to $k$.

The numbers that satisfy up to 40000 are:

1, 2, 10, 22, 58, 62, 63, 64, 66, 67, 68, 118, 178, 418, 838, 1258, 1264, 1265, 1277, 1278, 1678, 2098, 4618, 9238, 10508, 10509, 10510, 10512, 10513, 10514, 13858, 14704, 14754, 18478, 23098, 23102, 23276, 27718

Table 1 shows the numbers from OEIS A333703 together with their primorial base equivalents and the progressive totals of the digits of the all the primorial numbers up and including each number. The primorial base representation I've employed here uses the base 10 digits (0 to 9) together with a space as a separator (although colons are more commonly used). However, for numbers in the range up to 40000 that I use the base 12 system using the additional digits A for 10 and B for 11 are sufficient so that concatenation of the "placeholders" does not produce any ambiguity. The primorial number then looks like a normal base 12 number which produces an ambiguity in itself.

Table 1: permalink

Table 2 shows the numbers together with their corresponding progressive totals and the results when these totals are divided by the corresponing number.

Table 2: permalink

The next number after 22718 is 60058 so I won't be around to see that. For more information see this source. I started this blog by referring to my diurnal age on the 21st February 2025 (27718) but my diurnal age on the very next day ($\textbf{27719}$) also has a property that connects it to the primorial number base.

A343048   a($n$) is the least number whose sum of digits in primorial base equals $n$.

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

0, 1, 3, 5, 11, 17, 23, 29, 59, 89, 119, 149, 179, 209, 419, 629, 839, 1049, 1259, 1469, 1679, 1889, 2099, 2309, 4619, 6929, 9239, 11549, 13859, 16169, 18479, 20789, 23099, 25409, 27719, 30029

Table 3 shows the increasing values of $n$:

Table 3: permalink

Not the Sum of Distinct Squares or Cubes

I guess I'd never really thought about the issue before. I've written about what numbers can and cannot be expressed as a sum of two squares and what numbers cannot be expressed as a sum of three squares but what numbers $\textbf{cannot}$ be expressed a sum of two or more distinct squares? Well, the answer is not many and 128 is the largest of them. These number form OEIS A001422 :

A001422    Numbers which are $\textbf{not}$ the sum of $\textbf{distinct}$ squares.

The numbers are: 2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44, 47, 48, 60, 67, 72, 76, 92, 96, 108, 112, 128

This led me on to OEIS A001476 that deals with the same issue but involving cubes. 

A001476    Numbers that are $\textbf{not}$ the sum of $\textbf{distinct}$ positive cubes.

2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 66, 67, 68, 69, 70, 71, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 93,94, 95, 96, 97, 98

These are the initial terms below 100 and the OEIS comments go on to say the following:

There are 85 terms below 100, 793 terms below 1000, but only 2765 terms below 10000, and only 23 more up to the largest term a(2788)=12758.

An Interesting Triple 7 Number

Today I turned $\textbf{27717}$ days old and this number has a plethora of interesting properties that deserve a special mention and thus a dedicated post. Here are some of those properties.

• $\textbf{27717}$ is a so-called Lucky Cube, meaning it is a number whose cubes contain the digit sequence “888”, here: $$27717^3 = 21293088810813$$ The numbers that satisfy from 27717 to 40000 are: 27717, 27942, 27973, 28192, 28442, 28484, 28692, 28740, 28942, 29079, 29192, 29354, 29387, 29391, 29418, 29420, 29442, 29491, 29642, 29692, 29942, 29989.
  
  
• $\textbf{27717}$ is the lesser of a pair of adjacent composite numbers such that both are only one step away from their home primes. Here: 
• $27717 = 3 \times 9239 \rightarrow 39239$
• $27718 = 2 \times 13859 \rightarrow 213859$
  
  
• $\textbf{27717}$ is a number such that n + POD(n) and n - POD(n) are both prime (where POD stands for Product Of Digits). Here we have POD = 686:
• $27717 + 686 = 28403$ which is a prime number
• $27717 - 686 = 27031$ which is a prime number
  
  
• $\textbf{27717}$ is an interprime number because it is at equal distance from the previous prime (27701) and the next prime (27733).
  
  
• $\textbf{27717}$ is a number whose sum of divisors has prime factors (ignoring multiplicity) that multiply to the factorial 2310 where
  
  $2310= 2 \times 3 \times 5 \times 7 \times 11$
  
  Here 27717 has a sum of divisors 36960 and
  
  $36960= 2^5 \times 3 \times 5 \times 7 \times 11$
  
  but also forms a consecutive pair with 27718 because its sum of the divisors is 41580 and
  
  $41580= 2^2 \times 3^3 \times 5  \times 7 \times11$
  
  See blog post Primorials and the Sigma Function.
  
  
• $\textbf{27717}$ is the TENTH member of an interesting number chain (which is base independent):
  
• $27708 = 12 \times 2309$
• $27709 = 11 \times 2519$
• $27710 = 10 \times 2771$
• $27711 = 9 \times 3079$
• $27712 = 8 \times 3464$
• $27713 = 7 \times 3959$
• $27714 = 6 \times 4619$
• $27715 = 5 \times 5543$
• $27716 = 4 \times 6929$
• $27717 = 3 \times 9239$
• $27718 = 2 \times 13859$
  

See blog post Count Down Number Chains  

 

• $\textbf{27717}$ is a cyclic number.
  
  
• $\textbf{27717}$ is a xenodrome in base 9 : 42016. See blog post Xenodromes.
  
  
• $\textbf{27717}$ is a number that does not reach a palindrome after 2001 cycles of the reverse and add algorithm.
  
  
• $\textbf{27717}$ is a D-number meaning it is a number $n > 3$ such that n divides $k^{n-2}- k$ for all $1 < k < n$ relatively prime to $n$.
  
  
• $\textbf{27717}$ can be rendered as a digit equation as follows: $2 - \dfrac{7}{7} = 1 ^ 7$

Special Primes

I'm surprised that I haven't written about these sorts of primes before. The primes form OEIS A092529:

A092529  primes $p$ such that both the digit sum of $p$ plus $p$ and the digit product of $p$ plus $p$ are also primes (zeroes are not permitted).

Up to 40000, there are 136 such primes and they are (permalink):

163, 233, 293, 431, 499, 563, 617, 743, 1423, 1483, 1489, 1867, 2273, 2543, 2633, 3449, 4211, 4217, 4273, 4547, 4729, 5861, 6121, 6529, 6637, 6653, 6761, 6857, 6949, 7681, 8273, 8431, 8837, 8839, 9649, 9689, 11251, 11657, 11677, 11897, 12379, 12553, 13163, 13457, 13523, 13697, 13729, 13877, 13879, 14423, 14533, 14537, 14957, 15121, 15217, 15277, 15361, 15413, 15451, 15619, 15727, 15859, 16319, 16427, 16993, 17183, 17299, 17837, 18229, 18287, 18517, 19381, 19457, 19583, 21379, 21467, 21577, 21737, 21751, 21977, 21991, 22123, 22259, 22369, 22549, 22921, 23117, 23269, 23599, 23719, 24499, 24527, 25111, 25153, 25577, 25771, 25847, 25913, 25997, 26251, 26699, 26927, 27427, 28181, 29153, 29179, 32173, 32687, 32957, 32971, 33413, 33547, 33581, 33587, 33769, 33851, 34313, 34667, 35251, 35257, 35323, 35521, 35569, 35831, 36229, 36469, 36559, 36919, 37321, 37369, 37547, 37871, 38351, 38959, 39161, 39521

Let's just check the first number in this sequence, 163. The digit sum is 10 and the digit product is 18. Now 163 + 10 = 173 which is prime and 163 + 18 = 181 is also prime.

Interestingly if we consider subtraction instead of addition then no such primes exist. That is to say that, even up to one million, there are no primes such that $p$ minus the digit sum of $p$ and $p$ minus the digit product of $p$ are also primes (with zeroes not permitted). I'm not sure why this is so.

We can thin the ranks of the above primes if we require that the sum and product of the squares of the digits also form primes when added to the original prime. Only four numbers such satisfy all these criteria in the range up to 40000. These are 1423, 13697, 14533 and 33413 (permalink). Let's examine the first of these numbers 1423. The sum of digits is 10 and the product is 24. The digits squared are 1, 16, 4 and 9 with a sum of 30 and a product of 576. Thus we have:

• 1423 + 10 = 1433 (prime)
• 1423 + 24 = 1447 (prime)
• 1423 + 30 = 1453 (prime)
• 1423 + 576 = 1999 (prime)

If we extend the range up to one million, there are 48 numbers that satisfy:

1423, 13697, 14533, 33413, 53419, 57529, 61991, 71569, 91129, 125789, 128153, 132527, 132679, 143477, 149161, 159463, 223423, 238649, 275929, 284831, 288493, 297613, 316343, 337261, 343639, 367819, 375227, 441797, 447791, 498733, 512521, 573829, 574969, 582937, 613673, 626723, 722333, 723923, 728681, 735283, 746533, 748883, 752273, 762539, 766531, 836917, 869951, 872959

To thin this sequence further, let's impose additional criteria, specifically that the number plus the sum of the cubes of the digits and the number plus the product of the cubes of the digits must be prime as well. Here are the numbers up to one million: 125789, 132527, 573829 and 752273. Let's look at the first of these numbers, 125789:

• 125789 + 32 = 125821 (prime)
• 125789 + 5040 = 130829 (prime)
• 125789 + 224 = 126013 (prime)
• 125789 + 25401600 = 25527389 (prime)
• 125789 + 1718 = 127507 (prime)
• 125789 + 128024064000 = 128024189789 (prime)

Notice how the product of the squares and cubes of the digits are larger than the number itself. If we remove the requirement that the initial number be prime then a few more numbers satisfy the criteria for digits, digits squared and digits cubed. These are 13969, 65821, 125789, 132527, 349789, 537881, 545123, 573829 and 752273 (primes are shown in red).

A Seeming Coincidence

A friend of mine who turns 49 this year (2025) was born (of course) in 1976. I noticed that I will turn 76 this year and that I was born in 1949. If we ignore the '19' in our years of birth then our years of birth can be written as '76 and '49. The "coincidence" then is that we have this '76 and 49 versus '49 and 76 switching of the numbers. 

As I explained to my friend:

Taken to the extreme, I will turn 99 in 2048 in the unlikely event that I make it that far. Someone born in 1999 will turn 49 in this same year and so we have '99 and 49 versus '49 and 99. It doesn't hold in my case for people born in 2000 and beyond.

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.