Embedded Primes Versus Prime Partitions

I noticed that the number associated with my diurnal age today (28323), while not prime, has digits that are all prime with the exception of 8 and it also has two other primes contained within it (283 and 23). I asked Gemini the following question:

28323 is a composite number with the property that it is a concatenation of prime numbers in various ways e.g. 283 || 23 and 2 || 83 || 23 where I’ve used the symbol || for concatenation. Write a program in SageMath that will identify all composite numbers in a given range that are concatenations of prime numbers in two or more different ways. The output should be a list of comma-separated numbers. The default range can be from 4 to 40000 and the upper and lower limits of the range should be easily adjustable. There should also be a switch so that the program can search for prime numbers with the same property, although the default should be set to composite. The number of suitable concatenations should be adjustable as well, although 2 is to be the default. Can a count of the suitable numbers be included in the output, underneath the list of numbers.

The program as requested is quite flexible. Below are the opening lines of code:

# --- Adjustable Parameters ---

LOWER_LIMIT = 4

UPPER_LIMIT = 40000

SEARCH_PRIMES = True  # False = search for composite numbers (default), True = search for primes

MIN_WAYS = 11         # Minimum number of valid prime concatenations required

# -----------------------------

Using these settings and thus searching for all primes in the range from 4 to 40000 that can be concatenated in at least 11 different ways, I found only a single number: 37337 (permalink). However, OEIS A179919 that lists primes with eleven embedded primes listed 111317 as the first number. Clearly I was confused and I sought clarification from Gemini. Here is its explanation:

The discrepancy comes down to two fundamentally different mathematical definitions: full-string prime partitions (concatenations) versus distinct proper prime substrings (embedded primes).

Your SageMath program counts the number of ways an entire number can be sliced into a complete concatenation of prime pieces from left to right, whereas OEIS A179919 counts how many distinct prime numbers appear anywhere inside the number as contiguous substrings (excluding the prime itself).

1. What Your Program Counts: Prime Concatenations (37337)

Your program requires every single digit of the number to be used in each partition. A 5-digit number has 2^{5-1} - 1 = 15 possible ways to be sliced into two or more contiguous pieces.

For the prime number 37337, 11 of those 15 full-string partitions consist entirely of prime numbers:

1. 3 || 7 || 3 || 3 || 7

2. 3 || 7 || 3 || 37

3. 3 || 7 || 337

4. 3 || 73 || 3 || 7

5. 3 || 73 || 37

6. 37 || 3 || 3 || 7

7. 37 || 3 || 37

8. 37 || 337

9. 373 || 3 || 7

10. 373 || 37

11. 3733 || 7

(The only 4 slices of 37337 that fail are the ones containing the composite substrings 33 or 733: 3 || 7 || 33 || 7, 3 || 733 || 7, 37 || 33 || 7, and 3 || 7337.)

2. What OEIS A179919 Counts: Distinct Embedded Primes (111317)

In the OEIS family of sequences (such as A039996 and A179909–A179924), an "embedded prime" is defined as a distinct prime proper substring occurring anywhere within the base-10 representation of the prime.

Under the OEIS definition, three completely different rules apply:

• The rest of the digits do not need to form primes: You just extract a contiguous block of digits from the number and ignore the digits to the left and right.

• Duplicates are only counted once: You collect the substrings into a mathematical set of unique prime values.

• Composite digits like 1 are helpful: Because you don't have to partition the entire number into primes, having the digit 1 (which is not prime and ruins full concatenations) is actually an advantage in A179919 because 1 starts many multi-digit primes (11, 13, 17, 113, 131, 317, etc.).

If we evaluate your number 37337 using the OEIS "embedded primes" rule, it only has 6 embedded primes:

$$\{3, 7, 37, 73, 337, 373, 3733\} \longrightarrow \text{7 primes?}$$

Actually, checking all contiguous proper substrings of 37337:

• 1-digit: 3, 7 (repeats ignored)

• 2-digit: 37, 73 (33 is composite)

• 3-digit: 373, 337 (733 is prime! Wait: 733 is indeed prime!)

• 4-digit: 3733 (prime), 7337 (7337 = 11 \times 667, composite)

Even with 733 being prime, the set of distinct proper prime substrings of 37337 is \{3, 7, 37, 73, 337, 373, 733, 3733\}, which is only 8 distinct embedded primes (falling short of 11). Furthermore, note that because 733 is prime, 37337 actually has 12 valid prime concatenations in your program (3 || 733 || 7 is also valid!), which is why it passed your MIN_WAYS = 11 threshold (>= 11).

Now compare that to 111317, the first term of OEIS A179919:

• Under your concatenation program: 111317 has only 1 valid prime concatenation (11 || 13 || 17), because any slice that isolates a single 1 (1 || 11317, 111 || 3 || 17, etc.) fails since 1 is not prime.

• Under OEIS A179919 (distinct proper prime substrings): Scanning all contiguous substrings of 111317 yields 11 distinct primes:

  1. 1-digit: 3, 7 (2 primes)

  2. 2-digit: 11, 13, 17, 31 (4 primes)

  3. 3-digit: 113, 131, 317 (3 primes)

  4. 4-digit: 1113 (3 \times 371, composite), 1131 (3 \times 377, composite), 1317 (3 \times 439, composite) (0 primes)

  5. 5-digit: 11131 (prime), 11317 (prime) (2 primes)

Summing those unique prime substrings gives 2 + 4 + 3 + 0 + 2 = 11 embedded primes:

$$\{3, 7, 11, 13, 17, 31, 113, 131, 317, 11131, 11317\}$$

Side-by-Side Comparison of the Two Rules

Property	Your Program (Prime Concatenations)	OEIS A179919 (Embedded Primes)
Core Mechanism	Partitions the entire string into adjacent prime blocks (A || B || C).	Extracts any contiguous prime substring from anywhere inside the string.
Unused Digits	Forbidden — every digit must belong to a prime block in the partition.	Allowed — surrounding digits outside the substring are ignored.
Repeated Primes	Counted every time they form a distinct full-string partition.	Ignored — only distinct prime values are counted.
Score for 37337	12 valid full-string prime concatenations.	8 distinct embedded primes (\{3, 7, 37, 73, 337, 373, 733, 3733\}).
Score for 111317	1 valid full-string prime concatenation (11 || 13 || 17).	11 distinct embedded primes (\{3, 7, 11, 13, 17, 31, 113, 131, 317, 11131, 11317\}).

I've gotten Gemini to write a program that accepts any positive integer as input, prime or non-prime, and outputs the concatenated primes as well as the embedded primes. Here is the output for 28323 (permalink):

Analysis for 28323 (Composite)

=============================================

1. PRIME CONCATENATIONS (Total: 4)

   1. 2 || 83 || 2 || 3

   2. 2 || 83 || 23

   3. 283 || 2 || 3

   4. 283 || 23

2. DISTINCT EMBEDDED PRIMES (Total: 5)

   2, 3, 23, 83, 283

I've added this program to my daily number analysis.

When Dates and Days Align

I put this little problem to Gemini:

A person is born on April 3rd 1949. The day that he is born is reckoned as Day 0 and for every subsequent day 1 is added and this count becomes his diurnal age. Let the number associated with any date be determined as DDMYY where DD is the number associated with the month(1 to a maximum of 31), M is the month (from 1 to 9) and YY is the last two digits of the current year. Thus a date of 14th September 2026 would become 14926. Is there any date, using this system, where the diurnal age equals the date?

Here was its response:

September 23, 2014, is the sole date where the diurnal age mathematically equates to its DDMYY numerical representation. 

That is interesting and so, on the 23rd of September 2014, I was 23914 days old. I got Gemini to write a program that would accept any birthdate as input using this prompt:

Can you write a program in SageMath that will accept any birthdate as input and from there calculate if there is a date where the diurnal age equals the date number using DDMYY format. Let’s realistically limit the person’s possible age to less than 100.  

Here is a permalink to the program and the result for the following birthdate:

• 23rd August 1964:
  Match found: 05 February 1979 | Diurnal Age / Date Number: 5279

However, I realised that I could be less restrictive with the date format and modified the prompt as follows:

I’d like to modify the program so the acceptable date format can be DMMYY, DDMYY or DMYY. This ensures that the resultant number derived from the date is always less than six digits. Everything else remains the same.

Here is a permalink to the modified program which produces the same results for the two dates examined above. Here are some additional dates:

• 23rd November 1980:
  Match found: 06 July 1999 | Diurnal Age: 6799 | Date Format: 6799
  Match found: 16 January 2025 | Diurnal Age: 16125 | Date Format: 16125
  
  
• 6th October 2002:
  Match found: 10 September 2032 | Diurnal Age: 10932 | Date Format: 10932

So just a little curiosity. Not all birthdates yield a result within the span of 100 years. You could ask the person born on the 6th October 2002: how many days old will you be on the 10th September 2032? Don't know? Here's a way to find out:

• 10th \( \rightarrow\) 10
• September \( \rightarrow\) 9
• 2032 \( \rightarrow\)  32
• Concatenation \( \rightarrow\) 10932 \( \rightarrow\) your diurnal age!

Supercharged Semiprimes

 There is a set of non-square semiprimes defined by two criteria:

• the two prime factors concatenate in either order to form two new semiprimes
• the digits of the two prime factors can each be rearranged to form semiprimes

28291 is an example of such a semiprime. Let's examine its properties where || represents concatenation:$$ \begin{align} 28291 &=19 \times 1489\\ 19 \, || \, 1489 &= 191489 \\ &= 53 \times 3613 \\ 1489 \, || \,19  &=148919 \\ &= 137 \times 1087 \\ 19 &\rightarrow 91 \\&=7 \times 13 \\1489 &\rightarrow 1894 \\ &= 2 \times 947 \end{align}$$Up to 40000, the semiprimes that meet both criteria are (permalink):

779, 817, 1121, 1763, 1957, 2071, 2869, 3173, 3403, 3629, 4579, 4687, 4897, 5149, 5977, 6973, 7009, 7181, 7261, 7367, 7493, 8077, 8341, 8413, 8549, 8851, 8977, 9071, 9937, 10123, 10363, 10393, 10679, 10697, 10951, 11051, 11419, 11647, 12407, 12599, 13207, 14317, 14473, 14719, 14809, 14953, 15007, 15143, 15529, 15637, 15751, 16283, 16297, 16321, 16351, 16789, 16873, 17671, 18391, 18563, 18643, 19177, 19247, 19451, 20081, 20311, 20653, 20729, 20989, 21071, 21223, 21733, 21829, 21887, 21971, 22313, 22361, 22489, 22819, 22837, 22879, 22987, 23083, 23351, 23521, 24257, 24377, 24559, 24751, 24757, 24823, 25061, 25843, 25901, 26179, 26239, 26617, 26671, 26969, 27089, 27161, 27199, 27331, 27383, 27589, 27913, 28177, 28291, 28801, 28907, 28937, 28999, 29149, 29329, 29621, 30301, 30571, 30847, 31111, 31921, 32101, 32239, 32293, 32387, 32651, 32699, 33307, 33907, 33991, 34093, 34571, 34579, 34633, 34873, 34927, 35137, 35209, 35341, 35389, 35587, 35701, 35881, 36031, 36199, 36689, 37069, 37127, 37867, 37901, 38021, 38141, 38173, 38323, 38477, 38497, 38989, 39187, 39433, 39707, 39757

Biprime and Triprime Chains

I noticed that the number 28273 has an interesting property. It is a triprime but its prime factors, when concatenated in ascending order, also form a triprime. The process can be repeated one more time. See the table below (permalink):

Step |          Number | Factorization
--------------------------------------------------
    1 |           28273 | 7^2 * 577
    2 |           77577 | 3 * 19 * 1361
    3 |         3191361 | 3 * 37 * 28751

This got me thinking about what numbers lead to record chains. I put Gemini to work and this is what it came up with in the range up to one million (permalink):

    Number |    Chain Length
----------------------------
         8 |               3
        44 |               5
      7685 |               8
     15831 |               9
    261291 |              10
    768932 |              11

As can be seen:

• a chain of length 3 is reached before  there is a chain of length 2
• a chain of length 5 is reached before there is chain of length 4
• a chain of length 8 is reached before a chain of 6 or 7.

Let's examine 15831 from the above list and see what it's chain looks like (permalink).

Step |          Number | Factorization
--------------------------------------------------
    1 |           15831 | 3^2 * 1759
    2 |          331759 | 19^2 * 919
    3 |         1919919 | 3 * 59 * 10847
    4 |        35910847 | 7 * 103 * 49807
    5 |       710349807 | 3 * 271 * 873739
    6 |      3271873739 | 19 * 191 * 901591
    7 |     19191901591 | 37 * 701 * 739943
    8 |     37701739943 | 7 * 73 * 73780313
    9 |     77373780313 | 19 * 487 * 8362021

The same thing can be done for biprimes. The record lengths up to one million are as shown:

    Number |    Chain Length
----------------------------
         4 |               2
        10 |               4
       161 |               6
      1126 |               7
      1253 |               9
    100462 |              11

As can be seen:

• a chain of length 4 is reached before a chain of length 3
• a chain of length 6 is reached before there is a chain of length 5
• a chain of length 9 is reached before there is a chain of length 8
• a chain of length 11 is reached before there is a chain of length10

Let's examine 1253 from the above table.

 Step |          Number | Factorization
--------------------------------------------------
    1 |            1253 | 7 * 179
    2 |            7179 | 3 * 2393
    3 |           32393 | 29 * 1117
    4 |          291117 | 3 * 97039
    5 |          397039 | 29 * 13691
    6 |         2913691 | 11 * 264881
    7 |        11264881 | 1231 * 9151
    8 |        12319151 | 13 * 947627
    9 |        13947627 | 3 * 4649209

I've incorporated this analysis of biprimes and triprimes into my daily number analysis. Note that the biprimes can be square numbers e.g. \(49=7^2\) and the triprimes can contain repeated factors or even be cubic numbers e.g. \(44 = 2^2 \times 11\) or \(27 = 3^3\). Biprimes with no repeated factors are referred to as square-free biprimes while triprimes with no repeated factors are referred to as sphenic numbers.

There's room for extra investigation of course. I've only considered concatenation of prime factors in ascending order. Concatenations in any order could be considered. I've also only listed the record breakers as they first appear. For example, with the triprimes a chain of length 3 is reached by the number 8 before a chain of length 2 is reached. Thus I could consider what numbers first reach a given length. I'll consider these options in a future post.

Super Sphenic Numbers

The number associated with my diurnal age today (28262) is what might be termed a "super sphenic number" as I'll explain in this post. Firstly however, its factorisation:$$28262=2 \times 13 \times 1087$$If we reverse its digits, we get the number 26282 and this number is also sphenic:$$26282 = 2 \times 17 \times 773$$Let's now concatenate the factors of 28262 in ascending order. This gives us the number 2131087 which is also sphenic:$$2131087 = 7 \times 167 \times 1823$$28262 has a sum of digits of 20 and if we add this to the original number we get the palindromic number 28282 which is also sphenic:$$28262+20=28282 = 2 \times 79 \times 179$$The number has a product of digits of 384 and if we subtract this from the original number we get 27878 which is sphenic:$$28262 - 384 = 27878 = 2 \times 53 \times 263$$If we consider only the internal digits of 28262, we get the number 826 which is also sphenic:$$2\, 826 \, 2 \rightarrow826=2 \times 7 \times 59$$When a sphenic number is considered as a sphenic brick then it has an associated number in the form of the brick's surface area. In the case of 28262, this associated surface area is 32662 square units and this number too is sphenic:$$32662 = 2 \times 7 \times 2333$$28262 has a sum of proper divisors that is also sphenic:$$ \sigma(28262) - 28262 =17434 = 2 \times 23 \times 379 $$The number has a sum of prime factors (1102) that is sphenic:$$2 +13+1087=1102 = 2 \times 19 \times 29$$28262 has a totient of 13032 which is not sphenic but its cototient (number - totient) of 15230 is:$$28262 - 13032 = 15230 = 2 \times 5 \times 1523$$So we can see that 28262 may well be termed a super sphenic number because of the above associations.

28241: Analysis of an Interesting Number

My diurnal age today, 28241, has a number of interesting properties and a couple of them I've not met before. Let's examine them one by one.

FIRST INTERESTING PROPERTY

28241 can be formed by a concatenation of powers of 2 since all its digits are powers of 2 but the digits of its prime factors are all powers of 3:$$ \begin{align} 28241 & \equiv 2^1 \, || \, 2^3 \, || \, 2^1 \, || \, 2^2 \, || \, 2^0 \\ 28241 &= 31 \times 911 \\ &\equiv 3^1 \, || \, 3^0 \times 3^2 \, || \, 3^0 \, || \, 3^0 \end{align}$$In the range up to 40000, this doesn't happen often (ignoring 1 and 121 whose factors contain only the digits 1):$$ \begin{align} 81 &= 3^4  \\ 1441 &= 11 \times 131 \\1881 &= 3^2 \times 11 \times 19 \\ 21241 &= 11 \times 1931 \\ 28241 &= 31 \times 911 \end{align}$$SECOND INTERESTING PROPERTY

28241 is a member of OEIS A227942:

A227942: semiprimes formed by inserting a semiprime between the semiprime's ordered factors.

In the range up to 40000, only the following numbers qualify:

393, 2105, 2147, 3155, 5255, 5357, 25829, 26231, 28241, 29447, 33913, 35719, 39331$$ \begin{align} 3 \textbf{9} 3 &= 3 \times 131 \\ 2 \textbf{10} 5 &= 5 \times 421 \\ 2 \textbf{14} 7 &= 19 \times 113 \\ 3 \textbf{15} 5 &= 5 \times 631 \\ 5 \textbf{25} 5 &= 5 \times 1051 \\ 5 \textbf{35} 7 &= 11 \times 487 \\ 2 \textbf{58} 29 &= 23 \times 1123 \\ 2 \textbf{62} 31 &= 17 \times 1543 \\ 2 \textbf{82} 41 &= 31 \times 911 \\ 2 \textbf{94} 47 &= 11 \times 2677 \\ 3 \textbf{39} 13 &= 11 \times 3083 \\ 3 \textbf{57} 19 &= 23 \times 1553 \\ 3 \textbf{93} 31 &= 37 \times 1063 \end{align}$$THIRD INTERESTING PROPERTY

28241 is a member of OEIS A367337:

A367337: member of the commas sequence  starting with 5.

I'm familiar with the commas sequences and have written about them in a post titled The Commas Sequence in December of 2023. I won't show the full sequence but this snippet gives an idea of what's going on:$$ \dots, 28077, 28149, 28241, 28253, 28285, \dots $$Up to 40000, the remaining members of the sequence are:

28253, 28285, 28337, 28409, 28501, 28513, 28545, 28597, 28669, 28761, 28773, 28805, 28857, 28929, 29021, 29033, 29065, 29117, 29189, 29281, 29293, 29325, 29377, 29449, 29541, 29553, 29585, 29637, 29709, 29801, 29813, 29845, 29897, 29969, 30062, 30085, 30138, 30221, 30234, 30277, 30350, 30353, 30386, 30449, 30542, 30565, 30618, 30701, 30714, 30757, 30830, 30833, 30866, 30929, 31022, 31045, 31098, 31181, 31194, 31237, 31310, 31313, 31346, 31409, 31502, 31525, 31578, 31661, 31674, 31717, 31790, 31793, 31826, 31889, 31982, 32005, 32058, 32141, 32154, 32197, 32270, 32273, 32306, 32369, 32462, 32485, 32538, 32621, 32634, 32677, 32750, 32753, 32786, 32849, 32942, 32965, 33018, 33101, 33114, 33157, 33230, 33233, 33266, 33329, 33422, 33445, 33498, 33581, 33594, 33637, 33710, 33713, 33746, 33809, 33902, 33925, 33978, 34061, 34074, 34117, 34190, 34193, 34226, 34289, 34382, 34405, 34458, 34541, 34554, 34597, 34670, 34673, 34706, 34769, 34862, 34885, 34938, 35021, 35034, 35077, 35150, 35153, 35186, 35249, 35342, 35365, 35418, 35501, 35514, 35557, 35630, 35633, 35666, 35729, 35822, 35845, 35898, 35981, 35994, 36037, 36110, 36113, 36146, 36209, 36302, 36325, 36378, 36461, 36474, 36517, 36590, 36593, 36626, 36689, 36782, 36805, 36858, 36941, 36954, 36997, 37070, 37073, 37106, 37169, 37262, 37285, 37338, 37421, 37434, 37477, 37550, 37553, 37586, 37649, 37742, 37765, 37818, 37901, 37914, 37957, 38030, 38033, 38066, 38129, 38222, 38245, 38298, 38381, 38394, 38437, 38510, 38513, 38546, 38609, 38702, 38725, 38778, 38861, 38874, 38917, 38990, 38993, 39026, 39089, 39182, 39205, 39258, 39341, 39354, 39397

FOURTH INTERESTING PROPERTY

28241 has the property that:

• sum of digits ( \(  \, \text{SOD}  \, \)  ) is prime: \(2 + 8 + 2 + 4 + 1 =17 \)
  
  
• sum of digits squared (  \( \, \text{SOD}^2 \, \) ) is prime: \(2^2 + 8^2 + 2^2 + 4^2 + 1^2 =89 \)
  
  
• sum of digits cubed ( \( \, \text{SOD}^3 \, \) ) is prime:  \(2^3 + 8^3 + 2^3 + 4^3 + 1^3 = 593 \)

This property is relatively common but nonetheless interesting.

28224: An Interesting Number

28224 has 194 entries in the OEIS which is extraordinarily high for a five digit number. In this post I'll be discussing some of this number's most interesting properties but not all of them. There are just too many. It's prime factorisation is:$$28224=2^6 \times 3^2 \times 7^2$$FIRST INTERESTING PROPERTY

Numbers that are perfect squares are quite rare in the range up to 40000. There are only 200 of them and 28224, my diurnal age today, is one of them. It has the property that:$$28224=168^2$$The number of days between my experience of them is a little less than a year. There is a gap of exactly 365 days between \(183^2\) and \(182^2\) since:$$ \begin{align} 183^2-182^2 &= (183 + 182)(183-182) \\ &=365 \times 1 \\ &=365 \end{align}$$I'll be \(33124\) or \(182^2\) days when I'm over \(90\) years old so I may not get to experience the transition from this square to the next.

28224 is also a Loeschian number since it is equal to \(72^2+ 72 \times 120 + 120^2\).

28224 also has a product of digits (256) that is a perfect square since \(256=16^2\).

SECOND INTERESTING PROPERTY

Numbers that are the sum of two positive cubes are relatively rare in the range up to 40000. In fact, there are only 378 numbers in the range up to 40000 and 28224 is one of them because:$$28224=22^3 + 26^3$$These numbers form OEIS A004999.

THIRD INTERESTING NUMBER

Energetic numbers are numbers that can be broken into two or more substrings and expressed as a sum of (possibly different) positive powers of those substrings. They form OEIS  A055480. 28224 is one such number because:$$28224=28^3 + 2^{11} + 2^7 + 4^6$$I discuss this category of numbers in my blog post Energetic Numbers.

FOURTH INTERESTING NUMBER

Friedman numbers are positive integers which can be written in some non-trivial way using its own digits, together with the symbols + – × / ^ ( ) and concatenation. 28224 is one such number because:$$28224 = (2 + 82)^2 × 4$$It is said to be a "nice" Friedman number because the digits are in the same order as the number. These numbers are listed on my blog post Narcissistic, D-Powerfull and Friedman Numbers.

FIFTH INTERESTING NUMBER

28224 has the property that certain of its factors (not necessarily prime) can be arranged to form a palindrome. Specifically:$$2 \times 2 \times 2 \times 882 \times 2 \times 2 = 22288222$$I've written about these sorts of numbers in a post titled Why Is 313131 An Interesting Number?

SIXTH INTERESTING PROPERTY

28224 is a concatenation of powers of 2 since:$$28224= 2^1 \; || \; 2^3 \; || \; 2^1 \; || \; 2^1 \; || \; 2^2$$I've written about numbers that can be formed in this way in a blog post titled Nothing New Under The Sun. It is also a concatenation of multiples of 7 since:$$28224= (7 \times 4) \, || \, (7 \times 32)$$I posted about these sorts of concatenations in my blog post More Numbers as Concatenations.

SEVENTH INTERESTING PROPERTY

28224 is a member of OEIS A253824 where$$ \text{numbers } m = s \, || \, t \text{ such that } m = \sigma(s) \times \sigma(t)$$where || represents concatenation. In the case of 28224 we have:$$ \begin{align} 28224 &= 28 \, || \, 224 \\ &= \sigma(28) \times \sigma(224) \\ &= 56 \times 504 \\ &=28224 \end{align}$$28224 is only the third such number in the range up to 40000. The two earlier numbers are 540 and 2352.

EIGHTH INTERESTING PROPERTY

28224 has a digit sum of 18 and when this is added to the number the result is 28242 which has the same digits as 28224 but in a slightly different order. This property makes it a member of OEIS A246420.

A246420

Numbers \(n\) such that \(n\)  + digit sum of \(n\) is a permutation of the decimal digits of \(n\) .

28215: An Interesting Number

I've heard it said that all numbers are interesting and that, if a number is not, then it's interesting because it's not. The number associated with my diurnal age today (\( \textbf{28215}\) ) is definitely interesting. 

FIRST INTERESTING PROPERTY

Its prime factorisation is as follows:$$28215=3^3 \times 5 \times 11 \times 19$$It has 32 proper divisors and these are:

1, 3, 5, 9, 11, 15, 19, 27, 33, 45, 55, 57, 95, 99, 135, 165, 171, 209, 285, 297, 495, 513, 627, 855, 1045, 1485, 1881, 2565, 3135, 5643, 9405

The sum of these divisors is 29385 and so the number is abundant because this sum exceeds the number itself. Furthermore, all of these divisors are deficient and this makes it primitive abundant. Lastly the number is odd. This makes 28215 an odd primitive abundant number and these sorts of numbers are quite rare. Here is the list of the 50 such numbers up to 40000 (permalink):

945, 1575, 2205, 3465, 4095, 5355, 5775, 5985, 6435, 6825, 7245, 7425, 8085, 8415, 8925, 9135, 9555, 9765, 11655, 12705, 12915, 13545, 14805, 15015, 16695, 18585, 19215, 19635, 21105, 21945, 22365, 22995, 23205, 24885, 25935, 26145, 26565, 28035, \( \textbf{28215}\), 29835, 30555, 31395, 31815, 32445, 33345, 33495, 33915, 34155, 35805, 39585

SECOND INTERESTING PROPERTY

It can be seen from its factorisation that 28215 is 6-almost prime and its reversal, 51282, is also 6-almost prime:$$ \begin{align} 28215 &=3^3 \times 5 \times 11 \times 19 \\ 51282 &= 2 \times 3^2 \times 7 \times 11 \times 37 \end{align}$$While this property is not quite are rare as being odd primitive abundant, there are still only 118 such numbers in the range up to 40000. These are (permalink):

2576, 2772, 2970, 2992, 4284, 4356, 4410, 4600, 4698, 4824, 5265, 5625, 6534, 6752, 6776, 6900, 8008, 8250, 8964, 10710, 10890, 13140, 13986, 16236, 16335, 17577, 18504, 19494, 20286, 20574, 21112, 21114, 21150, 21160, 21336, 21492, 21576, 21609, 21712, 21900, 21912, 21996, 22392, 22770, 22788, 22824, 22869, 23058, 23247, 23250, 23496, 23562, 23580, 23598, 23632, 23832, 24156, 24660, 24975, 25020, 25092, 25104, 25164, 25245, 25300, 25416, 25434, 25452, 25608, 25668, 25752, 25952, 26163, 26334, 26532, 27060, 27108, 27135, 27192, 27240, 27248, 27270, 27405, 27408, 27468, 27472, 27588, 27608, 27636, 27816, 28116, \( \textbf{28215} \), 28314, 28710, 28782, 28890, 29052, 29172, 29322, 29340, 29392, 29412, 29580, 29750, 29784, 29835, 29900, 29960, 29984, 32967, 34965, 35775, 35937, 36162, 36990, 37026, 38367, 38934

I explore this reversibility extensively in my post titled Beyond Emirp from July 2025.

THIRD INTERESTING PROPERTY

The totient of a number \(n\) (commonly known as Euler's totient function or phi function, denoted as \(\phi(n)\)) counts the number of positive integers up to \(n\) that share no common factors with \(n\) other than 1. These numbers are called "relatively prime" or "coprime" to \(n\).

28215 has the property that it and its reversal, 51282, both have the same totient. Thus:$$ \phi(28215)=\phi(51282)=12960 $$This property is the rarest of all because in the range up to 40000 there are only 25 numbers with this property. They are:

190, 427, 429, 724, 924, 4147, 4697, 6276, 6726, 7414, 7964, 9079, 9709, 10040, 10940, 14450, 15860, 19190, 20493, 20553, 28092, \( \textbf{28215}\), 29082, 35502, 39402

FOURTH INTERESTING PROPERTY

28215 is a member of OEIS  A076773:

A076773   2-nadirs of \( \phi\) : numbers \( k \text{ such that }\)  

\( \phi(k-2) \gt \phi(k-1) \gt \phi(k) \lt \phi(k+1) \lt \phi(k+2) \)

I covered this in a post titled Totient Function: Jagged Versus Rounded Local Minima back in March of 2025. Figure 1 gives an idea of what is going on:

Figure 1

Here the \(\phi\) values for 28213, 28214, 26215, 28216, 28217 are:$$27808 \lt14106 \lt 12960 \lt14104 \lt23184$$There are 238 such minima in the range up to 40000:

315, 525, 735, 1155, 1365, 1575, 1755, 1785, 1815, 1995, 2145, 2415, 2475, 2805, 3045, 3315, 3465, 3885, 4095, 4125, 4305, 4515, 4725, 4935, 5115, 5145, 5355, 5775, 6045, 6195, 6405, 6435, 6615, 6825, 7035, 7095, 7245, 7395, 7455, 7605, 7665, 8085, 8265, 8505, 8715, 8745, 8925, 9135, 9345, 9405, 9555, 9735, 9765, 9975, 10185, 10395, 10455, 10545, 10815, 10965, 11055, 11235, 11385, 11445, 11655, 11865, 12075, 12285, 12495, 12675, 12705, 12915, 13125, 13335, 13545, 13695, 13965, 14025, 14175, 14355, 14385, 14595, 14805, 14835, 15015, 15045, 15225, 15405, 15435, 15645, 15675, 15855, 16005, 16065, 16275, 16335, 16485, 16695, 16905, 17085, 17325, 17355, 17745, 17955, 18135, 18165, 18375, 18585, 18645, 18795, 18975, 19215, 19425, 19635, 19665, 20055, 20265, 20295, 20475, 20625, 20685, 20865, 20895, 21105, 21255, 21315, 21525, 21945, 22365, 22425, 22575, 22605, 22785, 22995, 23205, 23265, 23415, 23595, 23625, 23655, 23835, 23985, 24225, 24255, 24675, 24885, 24915, 25095, 25245, 25305, 25515, 25575, 25725, 25905, 25935, 26145, 26325, 26565, 26775, 26985, 27027, 27195, 27615, 27825, 27885, 28035, \( \textbf{28215}\), 28245, 28275, 28455, 28665, 28815, 28875, 29055, 29295, 29505, 29865, 29925, 30195, 30345, 30555, 30723, 30765, 30975, 31185, 31365, 31395, 31605, 31815, 32025, 32175, 32235, 32445, 32655, 32835, 32895, 33033, 33075, 33345, 33495, 33705, 33735, 33915, 34125, 34155, 34335, 34485, 34515, 34545, 34755, 34965, 35175, 35385, 35805, 36225, 36435, 36465, 36645, 36795, 36855, 37065, 37275, 37455, 37485, 37695, 37905, 38115, 38535, 38745, 38775, 38955, 39165, 39195, 39375, 39435, 39585, 39765, 39795

FIFTH INTERESTING PROPERTY

28215 is also a member of A323380:

A323380  2-zeniths of \(\sigma\): numbers \( k \text{ such that }\)

                    \( \sigma(k-2) \lt \sigma(k-1) \lt \sigma(k) \gt \sigma(k+1) \gt \sigma(k+2) \)

I covered this in a post titled Totient and Sigma Graphs Revisited in August of 2025. Figure 2 shows what's going on and it's the local zenith replacing the local nadir of the totient graph:

Figure 2

Here the sigma values for 28213, 28214, 26215, 28216, 28217 are:$$28620 \lt 42324 \lt 57600 \gt52920 \gt 33600$$Below is a list of numbers up to 40000 that belong to BOTH OEIS A323380 and OEIS A076773 (sigma and totient respectively):

315, 525, 1155, 1575, 1755, 1785, 1995, 2475, 2805, 3045, 3315, 3465, 3885, 4095, 4125, 4515, 4725, 5115, 5355, 5775, 6045, 6195, 6405, 6435, 6615, 6825, 7035, 7245, 7605, 8085, 8505, 8715, 8925, 9135, 9405, 9555, 9765, 9975, 10395, 11235, 11385, 11445, 11655, 12075, 12285, 12675, 12705, 12915, 13125, 13545, 13965, 14025, 14175, 14355, 14595, 14805, 15015, 15435, 15645, 15675, 16005, 16065, 16275, 16335, 16695, 16905, 17325, 17745, 17955, 18135, 18375, 18585, 18795, 19215, 19635, 20475, 20685, 21105, 21315, 21525, 21945, 22365, 22605, 22995, 23205, 23595, 23625, 23835, 24255, 24675, 24885, 24915, 25245, 25515, 25725, 25935, 26325, 26565, 26775, 27027, 27195, 27885, 28035, \( \textbf{28215}\), 28245, 28275, 28665, 28875, 29295, 29925, 30195, 30345, 30555, 30723, 30765, 31185, 31365, 31395, 31605, 31815, 32025, 32175, 32235, 32445, 32835, 33075, 33345, 33495, 33915, 34125, 34155, 34485, 34515, 34755, 34965, 35175, 35805, 36225, 36435, 36645, 36795, 36855, 37275, 37485, 38115, 38745, 38955, 39165, 39195, 39375, 39435, 39585, 39765, 39795

SIXTH INTERESTING PROPERTY

28215 has a totient and sum of divisors that have 2, 3 and 5 as their distinct prime factors:$$ \begin{align} \sigma(28215) &= 57600 = 2^8 \times 3^2 \times 5^2 \rightarrow 2, 3, 5 \text{ as distinct prime factors} \\ \phi(28215) &= 12960 = 2^5 \times 3^4 \times 5 \rightarrow 2, 3, 5 \text{ as distinct prime factors} \end{align} $$There are 143 such numbers in the range from 28215 to 40000:

\( \textbf{28215} \), 28258, 28329, 28340, 28424, 28458, 28614, 28728, 28768, 28782, 28809, 28826, 28985, 29029, 29222, 29260, 29295, 29337, 29393, 29512, 29640, 29667, 29678, 29835, 29848, 30039, 30184, 30240, 30264, 30305, 30381, 30504, 30566, 30760, 30780, 30814, 30888, 30914, 30943, 30956, 30996, 31008, 31027, 31160, 31174, 31283, 31331, 31392, 31416, 31465, 31496, 31529, 31806, 31816, 32103, 32130, 32131, 32298, 32376, 32395, 32589, 32604, 32718, 32802, 32984, 33015, 33176, 33292, 33345, 33383, 33440, 33480, 33495, 33497, 33528, 33572, 33592, 33836, 33885, 33915, 34008, 34162, 34276, 34293, 34317, 34440, 34452, 34573, 34580, 34605, 34782, 34884, 35061, 35074, 35112, 35340, 35343, 35424, 35464, 35530, 35752, 35805, 35910, 35948, 35960, 36366, 36423, 36666, 36828, 36859, 36860, 36890, 36920, 37060, 37128, 37417, 37638, 37719, 37730, 37758, 37772, 37961, 38038, 38152, 38285, 38340, 38368, 38408, 38610, 38745, 38760, 38874, 39032, 39121, 39219, 39270, 39370, 39458, 39501, 39520, 39556, 39576, 39729

SEVENTH INTERESTING PROPERTY

28215 is what I've termed an \(a,b,c,d\) number because it can be combined with three other numbers, all with the same digits, to form a simple additive equation and this can be done in two different ways. Here is what I mean:$$ \begin{align} 25182 + 28125 + \textbf{28215} &= 81522 \\ 25812 + 28125 + \textbf{28215} &= 82152 \end{align}$$Here are the numbers with this property in the range from 28215 to 40000:

\( \textbf{28215}\), 28260, 28269, 28359, 28413, 28458, 28467, 28476, 28512, 28521, 28539, 28548, 28593, 28611, 28647, 28674, 28692, 28701, 28710, 28719, 28746, 28764, 28791, 28845, 28854, 28863, 28917, 28935, 28953, 28962, 28971, 29016, 29034, 29043, 29061, 29106, 29160, 29178, 29187, 29268, 29304, 29340, 29358, 29367, 29385, 29394, 29439, 29448, 29475, 29493, 29538, 29583, 29601, 29610, 29628, 29637, 29673, 29682, 29718, 29754, 29763, 29781, 29817, 29835, 29853, 29871, 29961, 30015, 30150, 30159, 30168, 30195, 30285, 30294, 30429, 30492, 30519, 30582, 30591, 30627, 30681, 30726, 30825, 30852, 30924, 30942, 30951, 31059, 31068, 31149, 31158, 31176, 31185, 31464, 31491, 31509, 31590, 31599, 31608, 31635, 31644, 31653, 31680, 31689, 31698, 31761, 31788, 31806, 31815, 31842, 31860, 31869, 31878, 31896, 31905, 31959, 31968, 31986, 31995, 32049, 32076, 32085, 32148, 32418, 32481, 32490, 32499, 32580, 32607, 32679, 32697, 32760, 32769, 32796, 32814, 32841, 32850, 32859, 32886, 32895, 32904, 32958, 32967, 32976, 32985, 32994, 34029, 34119, 34128, 34164, 34182, 34218, 34281, 34299, 34461, 34614, 34641, 34812, 34821, 34911, 34992, 35019, 35082, 35091, 35109, 35118, 35190, 35217, 35271, 35631, 35712, 35721, 35802, 35820, 35829, 35892, 35910, 35982, 35991, 36018, 36108, 36117, 36135, 36144, 36153, 36171, 36198, 36261, 36279, 36288, 36297, 36315, 36351, 36414, 36513, 36531, 36621, 36711, 36729, 36792, 36810, 36819, 36918, 36927, 36972, 36981, 37116, 37125, 37161, 37179, 37197, 37215, 37251, 37269, 37296, 37521, 37611, 37629, 37719, 37917, 37962, 38016, 38061, 38106, 38115, 38142, 38151, 38160, 38169, 38187, 38196, 38214, 38241, 38286, 38295, 38412, 38511, 38529, 38592, 38610, 38619, 38682, 38691, 38817, 38826, 38871, 38916, 38925, 38952, 38961, 39015, 39024, 39042, 39051, 39105, 39150, 39159, 39177, 39186, 39195, 39204, 39258, 39285, 39402, 39411, 39420, 39492, 39501, 39510, 39528, 39582, 39591, 39618, 39627, 39672, 39681, 39717, 39726, 39762, 39816, 39825, 39852, 39861, 39942, 39951

EIGHTH INTERESTING PROPERTY

If the prime factors, with multiplicity, of 28215 are concatenated in ascending order, they form a prime number. Thus:$$28215 =3^3 \times 5 \times 11 \times 19 \rightarrow 33351119$$Such a prime is called the home prime and so 28215 is only one step removed from its home prime. There are many other concatenations that yield primes and all the possibilities are listed below (permalink):

11193353, 11319533, 11335193, 19331153, 19335311, 19351133, 19511333, 31119353, 31131953, 31133519, 31153193, 31933511, 31951133, 31953113, 33191153, 33195311, 33311519, \( \textbf{33351119} \), 35113193, 35191133, 35193311, 35319113, 35331119, 35331911, 51131933, 51133193

This property, of being one step removed from its home prime, is relatively common but nonetheless interesting.

NINTH INTERESTING PROPERTY

28215 is a member of the commas sequence beginning with 8. I explore sequences of this type in my blog post The Commas Sequence from December of 2023. The full trajectory up to 40000 is as follows:

[8, 97, 168, 250, 252, 274, 317, 390, 393, 427, 502, 527, 603, 639, 736, 804, 852, 880, 888, 977, 1048, 1129, 1220, 1221, 1232, 1253, 1284, 1325, 1376, 1437, 1508, 1589, 1680, 1681, 1692, 1713, 1744, 1785, 1836, 1897, 1968, 2050, 2052, 2074, 2116, 2178, 2260, 2262, 2284, 2326, 2388, 2470, 2472, 2494, 2536, 2598, 2680, 2682, 2704, 2746, 2808, 2890, 2892, 2914, 2956, 3019, 3112, 3135, 3188, 3271, 3284, 3327, 3400, 3403, 3436, 3499, 3592, 3615, 3668, 3751, 3764, 3807, 3880, 3883, 3916, 3979, 4073, 4107, 4181, 4195, 4249, 4343, 4377, 4451, 4465, 4519, 4613, 4647, 4721, 4735, 4789, 4883, 4917, 4991, 5006, 5071, 5086, 5151, 5166, 5231, 5246, 5311, 5326, 5391, 5406, 5471, 5486, 5551, 5566, 5631, 5646, 5711, 5726, 5791, 5806, 5871, 5886, 5951, 5966, 6032, 6058, 6144, 6190, 6196, 6262, 6288, 6374, 6420, 6426, 6492, 6518, 6604, 6650, 6656, 6722, 6748, 6834, 6880, 6886, 6952, 6978, 7065, 7122, 7149, 7246, 7313, 7350, 7357, 7434, 7481, 7498, 7585, 7642, 7669, 7766, 7833, 7870, 7877, 7954, 8002, 8030, 8038, 8126, 8194, 8242, 8270, 8278, 8366, 8434, 8482, 8510, 8518, 8606, 8674, 8722, 8750, 8758, 8846, 8914, 8962, 8990, 8998, 9087, 9166, 9235, 9294, 9343, 9382, 9411, 9430, 9439, 9538, 9627, 9706, 9775, 9834, 9883, 9922, 9951, 9970, 9979, 10070, 10071, 10082, 10103, 10134, 10175, 10226, 10287, 10358, 10439, 10530, 10531, 10542, 10563, 10594, 10635, 10686, 10747, 10818, 10899, 10990, 10991, 11002, 11023, 11054, 11095, 11146, 11207, 11278, 11359, 11450, 11451, 11462, 11483, 11514, 11555, 11606, 11667, 11738, 11819, 11910, 11911, 11922, 11943, 11974, 12015, 12066, 12127, 12198, 12279, 12370, 12371, 12382, 12403, 12434, 12475, 12526, 12587, 12658, 12739, 12830, 12831, 12842, 12863, 12894, 12935, 12986, 13047, 13118, 13199, 13290, 13291, 13302, 13323, 13354, 13395, 13446, 13507, 13578, 13659, 13750, 13751, 13762, 13783, 13814, 13855, 13906, 13967, 14038, 14119, 14210, 14211, 14222, 14243, 14274, 14315, 14366, 14427, 14498, 14579, 14670, 14671, 14682, 14703, 14734, 14775, 14826, 14887, 14958, 15039, 15130, 15131, 15142, 15163, 15194, 15235, 15286, 15347, 15418, 15499, 15590, 15591, 15602, 15623, 15654, 15695, 15746, 15807, 15878, 15959, 16050, 16051, 16062, 16083, 16114, 16155, 16206, 16267, 16338, 16419, 16510, 16511, 16522, 16543, 16574, 16615, 16666, 16727, 16798, 16879, 16970, 16971, 16982, 17003, 17034, 17075, 17126, 17187, 17258, 17339, 17430, 17431, 17442, 17463, 17494, 17535, 17586, 17647, 17718, 17799, 17890, 17891, 17902, 17923, 17954, 17995, 18046, 18107, 18178, 18259, 18350, 18351, 18362, 18383, 18414, 18455, 18506, 18567, 18638, 18719, 18810, 18811, 18822, 18843, 18874, 18915, 18966, 19027, 19098, 19179, 19270, 19271, 19282, 19303, 19334, 19375, 19426, 19487, 19558, 19639, 19730, 19731, 19742, 19763, 19794, 19835, 19886, 19947, 20019, 20111, 20123, 20155, 20207, 20279, 20371, 20383, 20415, 20467, 20539, 20631, 20643, 20675, 20727, 20799, 20891, 20903, 20935, 20987, 21059, 21151, 21163, 21195, 21247, 21319, 21411, 21423, 21455, 21507, 21579, 21671, 21683, 21715, 21767, 21839, 21931, 21943, 21975, 22027, 22099, 22191, 22203, 22235, 22287, 22359, 22451, 22463, 22495, 22547, 22619, 22711, 22723, 22755, 22807, 22879, 22971, 22983, 23015, 23067, 23139, 23231, 23243, 23275, 23327, 23399, 23491, 23503, 23535, 23587, 23659, 23751, 23763, 23795, 23847, 23919, 24011, 24023, 24055, 24107, 24179, 24271, 24283, 24315, 24367, 24439, 24531, 24543, 24575, 24627, 24699, 24791, 24803, 24835, 24887, 24959, 25051, 25063, 25095, 25147, 25219, 25311, 25323, 25355, 25407, 25479, 25571, 25583, 25615, 25667, 25739, 25831, 25843, 25875, 25927, 25999, 26091, 26103, 26135, 26187, 26259, 26351, 26363, 26395, 26447, 26519, 26611, 26623, 26655, 26707, 26779, 26871, 26883, 26915, 26967, 27039, 27131, 27143, 27175, 27227, 27299, 27391, 27403, 27435, 27487, 27559, 27651, 27663, 27695, 27747, 27819, 27911, 27923, 27955, 28007, 28079, 28171, 28183, \( \textbf{28215} \), 28267, 28339, 28431, 28443, 28475, 28527, 28599, 28691, 28703, 28735, 28787, 28859, 28951, 28963, 28995, 29047, 29119, 29211, 29223, 29255, 29307, 29379, 29471, 29483, 29515, 29567, 29639, 29731, 29743, 29775, 29827, 29899, 29991, 30004, 30047, 30120, 30123, 30156, 30219, 30312, 30335, 30388, 30471, 30484, 30527, 30600, 30603, 30636, 30699, 30792, 30815, 30868, 30951, 30964, 31007, 31080, 31083, 31116, 31179, 31272, 31295, 31348, 31431, 31444, 31487, 31560, 31563, 31596, 31659, 31752, 31775, 31828, 31911, 31924, 31967, 32040, 32043, 32076, 32139, 32232, 32255, 32308, 32391, 32404, 32447, 32520, 32523, 32556, 32619, 32712, 32735, 32788, 32871, 32884, 32927, 33000, 33003, 33036, 33099, 33192, 33215, 33268, 33351, 33364, 33407, 33480, 33483, 33516, 33579, 33672, 33695, 33748, 33831, 33844, 33887, 33960, 33963, 33996, 34059, 34152, 34175, 34228, 34311, 34324, 34367, 34440, 34443, 34476, 34539, 34632, 34655, 34708, 34791, 34804, 34847, 34920, 34923, 34956, 35019, 35112, 35135, 35188, 35271, 35284, 35327, 35400, 35403, 35436, 35499, 35592, 35615, 35668, 35751, 35764, 35807, 35880, 35883, 35916, 35979, 36072, 36095, 36148, 36231, 36244, 36287, 36360, 36363, 36396, 36459, 36552, 36575, 36628, 36711, 36724, 36767, 36840, 36843, 36876, 36939, 37032, 37055, 37108, 37191, 37204, 37247, 37320, 37323, 37356, 37419, 37512, 37535, 37588, 37671, 37684, 37727, 37800, 37803, 37836, 37899, 37992, 38015, 38068, 38151, 38164, 38207, 38280, 38283, 38316, 38379, 38472, 38495, 38548

So, overall, 28215 is a very interesting number.

Forming Palindromes from Factors

 In a blog post titled Why Is 313131 An Interesting Number?, I remarked that:

$$ 313131=3 \times 7 \times 13 \times 31 \times 37$$If we rearrange the order of multiplication we get the following:$$ 313131=7 \times 3 \times 13 \times 31 \times 37$$Concatenating these digits we get the number \(73133137\) which is palindromic.

I went on to look at what other numbers in the range between 28000 and 29000 have this property and came up with the table shown below:

The numbers are thus relatively rare, there being only 27 in a range of 1000 numbers. This represents a density of 2.7%. The numbers are listed below:

28072, 28125, 28194, 28224, 28242, 28273, 28308, 28322, 28332, 28416, 28431, 28448, 28585, 28589, 28593, 28601, 28602, 28609, 28620, 28672, 28685, 28692, 28750, 28800, 28812, 28847, 28951

The factors under consideration here are all PRIME factors. What if we allow factors that are not necessarily prime. Take 28200 as an example:$$ \begin{align} 28200 &= 2 \times 5 \times 3 \times 2 \times 235 \times 2 \\ &\rightarrow 25322352 \end{align}$$The resultant number after concatenation of the factors is palindromic. Notice that the factor 235 is NOT prime.

It turns out that palindromes constructed in this way are relatively frequent. In the range between 28100 and 28300 (a range of only 200), the density is 15.4%. The numbers are:

28104, 28105, 28125, 28126, 28128, 28130, 28140, 28143, 28152, 28160, 28161, 28175, 28179, 28180, 28182, 28188, 28194, 28200, 28224, 28230, 28236, 28242, 28251, 28256, 28266, 28273, 28275, 28280, 28288, 28296, 28300

I've set up my multipurpose algorithm to identify such numbers when they pop up in my diurnal age analysis.

More Numbers as Concatenations

In my previous post, Divisors and Antidivisors: A Fresh Perspective, I dealt with numbers formed by concatenations of divisors and also by antidivisors of a number. For example, consider the number 15:

• 15 has divisors of 1, 3, 5 and 15
• 13515 is formed by concatenating these divisors
• 15 divides 13515 to give 901
• numbers with this property belong to OEIS A069872
• 15 has proper divisors of 1, 3 and 5
• 135 is formed by concatenating these proper divisors
• 15 divides 135 to give 9
• numbers with this property belong to OEIS A240265
• 15 has antidivisors of 2, 6 and 10
• 2610 is formed by concatenating these antidivisors
• 15 divides 2610 to give 174
• numbers with this property belong to OEIS A249764

In an earlier post titled, Nothing New Under The Sun, I looked at numbers formed by concatenation of powers of prime digits. Let's take 128864 which can be formed by a concatenation of powers of 2:$$128864=2^7 \, | \, 2^3 \, | \, 2^6$$where | is the symbol commonly used for concatenation. Numbers like this belong to OEIS A381259. 

In this post I want to look at numbers that are a concatenation of the multiples of a digit but that do not contain the digit itself. Let's take 28133 as an example. It's not immediately obvious but this number can be broke into two parts, 28 and 133, both of which are multiples of 7:$$28133 \rightarrow 28 \, | \, 133 = (7 \times 4) \, | \, (7 \times 19)$$There are 190 such numbers in the range up 40000. They are (permalink):

1414, 1421, 1428, 1435, 1442, 1449, 1456, 1463, 1484, 1491, 1498, 2121, 2128, 2135, 2142, 2149, 2156, 2163, 2184, 2191, 2198, 2828, 2835, 2842, 2849, 2856, 2863, 2884, 2891, 2898, 3535, 3542, 3549, 3556, 3563, 3584, 3591, 3598, 4242, 4249, 4256, 4263, 4284, 4291, 4298, 4949, 4956, 4963, 4984, 4991, 4998, 5656, 5663, 5684, 5691, 5698, 6363, 6384, 6391, 6398, 8484, 8491, 8498, 9191, 9198, 9898, 14105, 14112, 14119, 14126, 14133, 14140, 14154, 14161, 14168, 14182, 14189, 14196, 14203, 14210, 14224, 14231, 14238, 14245, 14252, 14259, 14266, 14280, 14294, 14301, 14308, 14315, 14322, 14329, 14336, 14343, 14350, 21105, 21112, 21119, 21126, 21133, 21140, 21154, 21161, 21168, 21182, 21189, 21196, 21203, 21210, 21224, 21231, 21238, 21245, 21252, 21259, 21266, 21280, 21294, 21301, 21308, 21315, 21322, 21329, 21336, 21343, 21350, 28105, 28112, 28119, 28126, 28133, 28140, 28154, 28161, 28168, 28182, 28189, 28196, 28203, 28210, 28224, 28231, 28238, 28245, 28252, 28259, 28266, 28280, 28294, 28301, 28308, 28315, 28322, 28329, 28336, 28343, 28350, 35105, 35112, 35119, 35126, 35133, 35140, 35154, 35161, 35168, 35182, 35189, 35196, 35203, 35210, 35224, 35231, 35238, 35245, 35252, 35259, 35266, 35280, 35294, 35301, 35308, 35315, 35322, 35329, 35336, 35343, 35350

Some of these numbers find themselves in the OEIS for reasons that involve multiples but in slightly different ways. Take 28168, listed above, as an example. It is a member of OEIS A009440: a(\(n\)) is the concatenation of \(n\) and 6\(n\). Thus we have:$$ \begin{align} 28168 &= 28 | (6 \times 28) \\28168 &= (7 \times 4) | (7 \times 24) \end{align}$$Choosing multiples of 2 and 3 produce 641 and 455 suitable numbers respectively while choosing multiples of 5 produces 78 suitable numbers in the range up to 40000 (permalink):

1010, 1020, 1030, 1040, 1060, 1070, 1080, 1090, 2020, 2030, 2040, 2060, 2070, 2080, 2090, 3030, 3040, 3060, 3070, 3080, 3090, 4040, 4060, 4070, 4080, 4090, 6060, 6070, 6080, 6090, 7070, 7080, 7090, 8080, 8090, 9090, 10100, 10110, 10120, 10130, 10140, 10160, 10170, 10180, 10190, 10200, 10210, 10220, 10230, 10240, 20100, 20110, 20120, 20130, 20140, 20160, 20170, 20180, 20190, 20200, 20210, 20220, 20230, 20240, 30100, 30110, 30120, 30130, 30140, 30160, 30170, 30180, 30190, 30200, 30210, 30220, 30230, 30240

In the case of 1010, we have:$$1010 \rightarrow 10 \, | \, 10 = (5 \times 2) \, | \, (5 \times 2)$$The permalink allows experimentation with other digits or even numbers. There's no deep Mathematics in all this just another way to spot patterns in numbers.

Divisors and Antidivisors: A Fresh Perspective

I've written about antidivisors before in posts titled Anti-Divisors and More on Anti-Divisors. Here is a fresh take. My diurnal age today is 28134 and the antidivisors of this number are 4, 12, 36, 108, 2084, 6252 and 18756. If we concatenate these numbers from left to right we get a rather large number (412361082084625218756) with the property that:$$ \frac{412361082084625218756}{28134} = 14657037111133334$$Numbers with this property form OEIS A249764:

 

A249764: numbers which divide the concatenation, in ascending order, of their anti-divisors.

Up to 40000, these numbers are relatively rare (permalink): 

15, 30, 105, 120, 150, 222, 375, 585, 1500, 1695, 1755, 1800, 2700, 3449, 3750, 3840, 4891, 6720, 7680, 12000, 13583, 14400, 15000, 18750, 19200, 20940, 28134, 30000, 34800, 35625

The OEIS lists an analog of this, namely OEIS A069872:

 

A069872: numbers \(k\) such that \(k\) divides the concatenation all divisors in ascending order.

Comparatively, members of this sequence are rather more numerous. There are 181 numbers in the range up to 40000 (permalink):

1, 2, 4, 5, 6, 8, 10, 15, 16, 20, 24, 25, 30, 32, 40, 50, 60, 64, 80, 90, 96, 100, 104, 120, 124, 125, 128, 150, 160, 200, 240, 250, 255, 256, 288, 320, 360, 375, 380, 384, 400, 425, 464, 480, 495, 500, 512, 600, 618, 625, 640, 750, 795, 800, 864, 875, 960, 1000, 1024, 1110, 1230, 1250, 1280, 1300, 1390, 1400, 1408, 1440, 1469, 1500, 1525, 1536, 1600, 1632, 1920, 2000, 2048, 2050, 2250, 2400, 2500, 2556, 2560, 2910, 2944, 2952, 3000, 3040, 3125, 3200, 3330, 3360, 3625, 3750, 3825, 3840, 4000, 4096, 4304, 4625, 4650, 4992, 5000, 5120, 5250, 5280, 5300, 5568, 5760, 6000, 6144, 6150, 6250, 6400, 6528, 6750, 7168, 7200, 7560, 7680, 8000, 8192, 9000, 9330, 9375, 9600, 10000, 10240, 10500, 10752, 11875, 12000, 12500, 12800, 13410, 13600, 14000, 15000, 15360, 15625, 15680, 16000, 16384, 16640, 17500, 17920, 18432, 18750, 19710, 20000, 20200, 20250, 20480, 20610, 21600, 21760, 22752, 23040, 23375, 24000, 24120, 24576, 25000, 25600, 25984, 27000, 28125, 30720, 31250, 32000, 32768, 33930, 34480, 34800, 35000, 36000, 37500, 37830, 38400, 39000, 40000

A variation on this is OEIS A240265:

A240265: numbers that divide the concatenation of their aliquot divisors, in ascending order.

These are less numerous. In the range up to 40000, they are:

4, 15, 16, 255, 375, 495, 795, 1469, 3825, 9375, 28125

All these numbers are also in OEIS A069872 and are marked in red in that sequence's list of members. It can be seen that 28125 is rather special and this number marked my diurnal age only 9 days ago. However, at the time, I missed its membership in these two sequences.

77th Birthday

Today I turned 77 years of age and my equivalent diurnal age is 28124 which has the following factorisation:$$28124=2 \times 2 \times 79 \times 89$$Though this number is composite, it has numerous prime number associations. Let's examine some of them beginning with its sum of digits, sum of digits squares and sum of digits cubed:$$ \begin{align} 2 + 8 + 1 + 2 + 4 &=17 \text{ (prime)} \\2^2+8^2+1^2+2^2+8^4 &= 89 \text{ (prime)} \\2^3+8^3+1^3+2^3+8^3 &= 593 \text{ (prime)} \end{align}$$The number is only one step removed from its home prime because:$$28124=2 \times 2 \times 79 \times 89 \rightarrow 227989 \text{ (prime)}$$The number is also a member of OEIS A048381: numbers such that replacing each nonzero digit with the n-th prime (replacing each 0 digit with a 1) yields a prime. Thus:$$28124 \rightarrow 319237 \text{ (prime)}$$The number has a binary complement that is prime. The binary complement of a number is determined by changing the number to binary and swapping any 0's for 1's and vice versa. Thus:$$ \begin{align} 28124_{10} &= 110110111011100_2 \\ &\rightarrow 001001000100011_2 \\ &=4643_{10} \text{ (prime)} \end{align}$$The number is quickly captured by the prime 28109 under the ODD(+) and EVEN(-) algorithm where the sum of the odd digits is added to the number and the sum of the even digits is subtracted recursively until a fixed point is reached or a loop is entered. Here is the trajectory is simply:$$ \begin{align} 28124 &\rightarrow 28124 + 1 -(2 + 8 + 2 + 4) \\ &=28124 + 1 - 16 \\ &=28109 \text{ (prime)} \end{align}$$The number can be considered as a concatenation of powers of the prime 2 because:$$ 28124 = 2^1\, | \,2^3 \,| \,2^0 \,| \,2^1 \,| \, 2^2 $$where | represents concatenation. The number can be generated by adding the prime sum (13) of the digits of the prime 28111 to itself. Thus:$$28111+13=28124$$The digits of the number can be rearranged to form the following primes:$$22481, 24281, 24821, 42281, 42821, 48221, 82241, 82421, 84221$$The position 28124 in the Recaman Sequence is reached after a prime number of iterations:$$0 \rightarrow 28124 \text{ requires } 34183 \text{ (prime) iterations}$$