28277: A Prime To Be Proud Of

Until today my diurnal age has suffered a drought of prime numbers. Prior to today (3rd September 2026) when I turned 28277 days old, the last prime (28229) occurred on July 17th. This marked a gap of 48 days between successive primes. This is not a record gap but it is impressive. Figure 1 shows the successive record gaps between primes.

Figure 1

I'll enumerate some of 28277's most interesting properties:

PROPERTY 1:

It forms a twin prime with 28279 but it also marks the beginning of gaps of 2, 4, 6, 8 and 10 between successive primes. The progression of primes is thus 28277, 28279, 28283, 28289, 28297, 28307. Such an occurrence is not common and membership is restricted to only three numbers (13901, 21557, 28277) in the range up to 40000. These and subsequent numbers constitute OEIS A190817.

PROPERTY 2

28277 is what is called a "good" prime and I posted about this type of prime in my blog post titled The Good Prime on the 12th of March 2025. As I explained there:

A prime \(p_n\) is said to be \( \textbf{good} \) if \(p_n^{^\textbf{2}}>p_{n-i } \cdot p_{n+i} \) for all \( 1 \leq i < n \).

The good primes from 28277 to 40000 are: 

28277, 28387, 28403, 28493, 28537, 28571, 28591, 28597, 29833, 29983, 30011, 30059, 30089, 30491, 30631, 30637, 30671, 30757, 30803, 31121, 31139, 31147, 31957, 32027, 32051, 32057, 32297, 32969, 33287, 33311, 33329, 34123, 35729, 35747, 35797, 35801, 35831, 35951, 35963, 36433, 36451, 36467, 36523, 36527, 36671, 38113, 38149, 38167, 38177, 38543, 38557, 38593, 38651, 38669, 39079, 39089

PROPERTY 3

28277 has what might be called an "internal prime". Strip away the first and last digits and what remains is 827, a prime number. I discuss these types of numbers in my post titled Numbers Within Numbers from the 21st June 2026. Primes with this property form OEIS A069686:

 A069686: primes whose internal digits form a prime.

From 28277 to 40000, the members of the sequence are:

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

PROPERTY 4

28277 gives prime 31931717 when digits become indices of prime numbers. Here we have:

• \(2 \rightarrow p_2=3\)
• \(8 \rightarrow p_8=19\)
• \(7 \rightarrow p_7=17\)

The primes of this sort from 28277 to 40000 are (permalink):

28277, 28289, 28319, 28429, 28807, 28979, 29137, 29347, 29399, 29717, 29819, 29837, 29917, 30059, 30089, 30169, 30187, 30367, 30467, 30469, 30497, 30509, 30689, 30697, 30707, 30727, 31139, 31159, 31177, 31247, 31277, 31337, 31469, 31489, 31687, 31847, 31849, 32099, 32309, 32569, 32717, 32957, 32987, 33049, 33469, 33577, 33629, 33797, 33809, 33889, 33937, 34019, 34129, 34259, 34297, 34327, 34439, 34469, 34607, 34649, 34807, 34819, 35027, 35159, 35407, 35597, 36037, 36107, 36109, 36467, 36469, 36587, 36637, 36809, 36857, 37087, 37139, 37199, 37277, 37337, 37579, 37889, 38189, 38299, 38327, 38459, 38609, 38639, 38699, 38747, 39047, 39119, 39239, 39397, 39667, 39779

PROPERTY 5

28277 is the average of a prime and its emirp in two different ways. The two ways are:

1. \( \dfrac{18773 + 37781}{2}= 28277\)
2. \( \dfrac{19763 + 36791}{2} = 28277\)

I discuss these sorts of primes in my blog post Prime Emirp Pair Averages of 12th May 2023. Such primes form OEIS A178587:

A178587

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

These primes are few and far between with the initial members being:

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

PROPERTY 6

28277 has the following trajectory under the Primes(+) and Non_Primes(-) algorithm: 

\(28277 \rightarrow 28287 \rightarrow 28282 \rightarrow 28272 \rightarrow 28277\)

It is thus a vortical and part of the vortex beginning and ending with 28277. This vortex has 36 captives.

Some Categories of Primes

There is a category of prime numbers with the property that when both the sum of their digits and the product of their digits is added to the number then the new, resultant numbers are also prime. An example would be 28181 with a sum of digits of 20 and a product of digits of 128 where:$$ \begin{align} 28181 + 20 &= 28201 \text{ prime} \\ 28181 + 128 &= 28309 \text{ prime} \end{align}$$In the range up to 40000, these primes have a density of 7.376% compared to all primes. Here is a list of such primes between 28000 and 40000 (permalink):

28097, 28181, 28703, 28901, 29153, 29179, 29209, 30089, 30119, 30203, 30313, 30449, 30469, 30539, 30557, 30649, 30661, 30713, 30803, 30809, 30829, 31019, 31307, 32063, 32069, 32083, 32173, 32203, 32401, 32687, 32957, 32971, 33013, 33037, 33091, 33301, 33413, 33547, 33581, 33587, 33769, 33851, 34313, 34667, 35053, 35059, 35251, 35257, 35323, 35507, 35509, 35521, 35569, 35831, 36209, 36229, 36469, 36559, 36607, 36919, 37019, 37039, 37097, 37321, 37369, 37501, 37507, 37547, 37871, 38047, 38351, 38959, 39019, 39079, 39103, 39161, 39301, 39521

These primes constitute OEIS A128717:

A128717: primes that yield another prime if one adds either the sum of its digits or the product of its digits.

Another category of prime involves its cube being pandigital, meaning that each digit from 0 to 9 occurs at least once with duplicates being permitted. Again 28181 satisfies this condition:$$28181^3 = 20753798525641$$Primes of this sort constitute:

A124629: primes \(p\) such that their cubes are pandigital.

The members of this sequence up to 40000 have a density is 1.523 % compared to all primes and these are (permalink):

5437, 6221, 7219, 8443, 10903, 11353, 15937, 17123, 18229, 19429, 20353, 20903, 20929, 21803, 21841, 21961, 22123, 22283, 22993, 23053, 23369, 23663, 24733, 25183, 25219, 25463, 26317, 26387, 26449, 27127, 27481, 28181, 28631, 28711, 28961, 29059, 29443, 29501, 30169, 31153, 31183, 32213, 32801, 33739, 33797, 33811, 33941, 34283, 35027, 35051, 35729, 35963, 36137, 36251, 36383, 36809, 36943, 37223, 37369, 37511, 37619, 37967, 38281, 38917

Another category of prime involves the average of the prime and the next prime being palindromic. Again 28181 satisfies since:$$ \frac{28181+28183}{2}=28182$$Many such primes are the lesser of a twin prime pair but not all. Primes of this sort constitute OEIS A242387:

A242387: lesser of consecutive primes whose average is a palindromic number.

The members of this sequence up to 40000 have a density of 1.213% compared to all primes and these are (permalink):

3, 5, 7, 97, 109, 281, 359, 389, 409, 509, 631, 653, 691, 743, 827, 857, 907, 937, 967, 1549, 2111, 2767, 4219, 4441, 7001, 9007, 9337, 9661, 10099, 11503, 12919, 13421, 16759, 17569, 21011, 21611, 23831, 26261, 26861, 28181, 29287, 29483, 30497, 31307, 32213, 33029, 33629, 34739, 36353, 37463, 39089

Another category of prime involves the differences between consecutive digits. Some primes have consecutive digits that differ by 6 or 7. An example is 28181 where we see that:$$ 2_{ \, 6} \, 8_{ \, 7} \, 1_{ \, 7} \, 8_{ \, 7} \, 1$$Such primes are few and far between and in the range up 40000, there are only the following:

17, 29, 71, 181, 281, 293, 607, 829, 929, 2939, 3929, 8171, 8293, 9281, 9293, 18181, 28181, 39293

Such primes belong to OEIS A048418:

A048418: primes whose consecutive digits differ by 6 or 7.

Yes another category involves totals of composite numbers between successive primes that are palindromes. 28181 qualifies once again because the next prime is its twin 28183 and the interprime number, 28182, is palindromic. Let's consider another prime, 29587. The next prime is 29599 and the composite numbers between them total 325523, a palindrome. Therefore we include 29587. These primes form OEIS A054266 with a density of only 0.8089% of the primes in the range up to 40000:

A054266: sum of composite numbers between prime \(p\) and nextprime(\(p\)) is palindromic.

The members up to 40000 are (permalink):

2, 3, 5, 109, 193, 281, 509, 661, 827, 857, 1439, 2111, 3433, 3889, 3967, 4549, 6661, 7001, 8467, 10099, 17203, 18583, 21011, 21611, 23831, 24847, 25117, 26261, 26497, 26861, 28181, 29587, 30497, 31307

We see that 28181, my diurnal age today, features in all these different categories of primes. Another category of primes (to which 28181 cannot belong) is to consider primes that only consist of non-prime digits (0, 1, 4, 6, 8 and 9). They do not contain any prime digits (2, 3, 5 or 7). Such primes belong to OEIS A034844 and comprise 5.782% of the primes up to 40000:

A034844: primes with only nonprime decimal digits.

Here are the primes up to 40000 (permalink):

11, 19, 41, 61, 89, 101, 109, 149, 181, 191, 199, 401, 409, 419, 449, 461, 491, 499, 601, 619, 641, 661, 691, 809, 811, 881, 911, 919, 941, 991, 1009, 1019, 1049, 1061, 1069, 1091, 1109, 1181, 1409, 1481, 1489, 1499, 1601, 1609, 1619, 1669, 1699, 1801, 1811, 1861, 1889, 1901, 1949, 1999, 4001, 4019, 4049, 4091, 4099, 4111, 4409, 4441, 4481, 4649, 4691, 4801, 4861, 4889, 4909, 4919, 4969, 4999, 6011, 6089, 6091, 6101, 6199, 6449, 6469, 6481, 6491, 6619, 6661, 6689, 6691, 6841, 6869, 6899, 6911, 6949, 6961, 6991, 8009, 8011, 8069, 8081, 8089, 8101, 8111, 8161, 8191, 8419, 8461, 8609, 8641, 8669, 8681, 8689, 8699, 8819, 8849, 8861, 8941, 8969, 8999, 9001, 9011, 9041, 9049, 9091, 9109, 9161, 9181, 9199, 9419, 9461, 9491, 9601, 9619, 9649, 9661, 9689, 9811, 9901, 9941, 9949, 10009, 10061, 10069, 10091, 10099, 10111, 10141, 10169, 10181, 10499, 10601, 10691, 10861, 10889, 10891, 10909, 10949, 11069, 11119, 11149, 11161, 11411, 11489, 11491, 11681, 11689, 11699, 11801, 11909, 11941, 11969, 11981, 14009, 14011, 14081, 14149, 14401, 14411, 14419, 14449, 14461, 14489, 14669, 14699, 14869, 14891, 14969, 16001, 16061, 16069, 16091, 16111, 16141, 16189, 16411, 16481, 16619, 16649, 16661, 16691, 16699, 16811, 16889, 16901, 16981, 18041, 18049, 18061, 18089, 18119, 18149, 18169, 18181, 18191, 18199, 18401, 18461, 18481, 18661, 18691, 18869, 18899, 18911, 18919, 19001, 19009, 19069, 19081, 19141, 19181, 19441, 19469, 19489, 19609, 19661, 19681, 19699, 19801, 19819, 19841, 19861, 19889, 19891, 19919, 19949, 19961, 19991

Primes beginning with 2 or 3 cannot qualify and so it is only when we reach primes beginning with 4 that membership is possible. The first of these is 40009.

We can flip this and consider only those primes that are comprised of prime digits. These form OEIS A019546:

A019546: primes whose digits are primes; primes having only {2, 3, 5, 7} as digits.

These primes have a density of 2.890% of the primes up to 40000 are they are (permalink):

2, 3, 5, 7, 23, 37, 53, 73, 223, 227, 233, 257, 277, 337, 353, 373, 523, 557, 577, 727, 733, 757, 773, 2237, 2273, 2333, 2357, 2377, 2557, 2753, 2777, 3253, 3257, 3323, 3373, 3527, 3533, 3557, 3727, 3733, 5227, 5233, 5237, 5273, 5323, 5333, 5527, 5557, 5573, 5737, 7237, 7253, 7333, 7523, 7537, 7573, 7577, 7723, 7727, 7753, 7757, 22273, 22277, 22573, 22727, 22777, 23227, 23327, 23333, 23357, 23537, 23557, 23753, 23773, 25237, 25253, 25357, 25373, 25523, 25537, 25577, 25733, 27253, 27277, 27337, 27527, 27733, 27737, 27773, 32233, 32237, 32257, 32323, 32327, 32353, 32377, 32533, 32537, 32573, 33223, 33353, 33377, 33533, 33577, 33757, 33773, 35227, 35257, 35323, 35327, 35353, 35527, 35533, 35537, 35573, 35753, 37223, 37253, 37273, 37277, 37337, 37357, 37537, 37573

Average of Three Primes Squared

The number associated with my diurnal age today is 28129 and it is a member of OEIS A075893:

     A075893: average of three successive primes squared with \(n > 3\) 
                

    \( \dfrac{\text{prime} (n)^2+ \text{prime}(n+1)^2+\text{prime}(n+2)^2)}{3}\)

The initial terms are:

65, 113, 193, 273, 393, 577, 777, 1057, 1337, 1633, 1913, 2289, 2833, 3337, 3897, 4417, 4953, 5537, 6153, 7017, 8073, 9177, 10073, 10753, 11313, 12033, 13593, 15353, 17353, 18417, 20097, 21441, 23217, 24673, 26369, 28129, 29953, 31577, 33761, 35497, 37513, 38553

The OEIS comments state that:

Unlike the average of three successive primes, the average of three successive primes (greater than 3) squared is always integral.

I asked Gemini why this was so:

The reason the average of the squares of three successive primes (greater than 3) is always an integer is based on a fascinating pattern about how primes behave when divided by 3.

In fact, this rule doesn't just apply to successive primes—it applies to any three primes greater than 3. Here is the step-by-step mathematical breakdown of why this happens.

1. All Primes Greater Than 3 Avoid Multiples of 3

By definition, a prime number is only divisible by 1 and itself. This means that any prime number strictly greater than 3 cannot be a multiple of 3.

If we divide any number by 3, we can only ever get one of three possible remainders: 0, 1, or 2.

• Because primes greater than 3 are not divisible by 3, they will never have a remainder of 0.
• Therefore, any prime number p > 3 must leave a remainder of either 1 or 2 when divided by 3.

In algebra, this means every prime p > 3 can be written in one of two ways, where k is an integer:

• p = 3k + 1 (leaves a remainder of 1)
• p = 3k + 2 (leaves a remainder of 2, which can also be written as p = 3k - 1)

2. Squaring the Primes

Now, let's look at what happens when we square these prime numbers.

Case A: The prime is 3k + 1
p² = (3k + 1)²
p² = 9k² + 6k + 1

We can factor out a 3 from the first two terms:
p² = 3(3k² + 2k) + 1

Because 3(3k² + 2k) is a multiple of 3, the entire expression leaves a remainder of 1 when divided by 3.

Case B: The prime is 3k - 1
p² = (3k - 1)²
p² = 9k² - 6k + 1

Again, factor out a 3 from the first two terms:
p² = 3(3k² - 2k) + 1

Just like the first case, this leaves a remainder of 1 when divided by 3.

The Key Rule: No matter which type of prime you have, as long as it is greater than 3, its square will always be exactly 1 more than a multiple of 3.

3. Averaging Three Squared Primes

Let p₁, p₂, and p₃ be any three primes greater than 3. Based on our rule above, we can express their squares as:

• p₁² = 3a + 1
• p₂² = 3b + 1
• p₃² = 3c + 1

Now, add them together to find the sum:
Sum = (3a + 1) + (3b + 1) + (3c + 1)
Sum = 3a + 3b + 3c + 3

We can factor out a 3 from the entire sum:
Sum = 3(a + b + c + 1)

Since the sum of the three squared primes is a perfect multiple of 3, dividing that sum by 3 to find the average will always result in a clean integer:
Average = 3(a + b + c + 1) / 3 = a + b + c + 1

Because a, b, and c are integers, the average itself is guaranteed to be an integer.

A Little Maths Puzzle

This puzzle appeared on PUZZLE A DAY and asks:

What is the only positive number that is equal to the average of its digits? Trailing zeros are not permitted. The trivial single digit answers don’t count as the question specifies ‘digits’. The clue provided is that the number consists of two digits. 

I made the mistake of thinking that the number is a two digit integer and reached an impasse. Let's represent the number with the digits \(x\) and \(y\). Then we have:$$ \begin{align} 10x + y &=\frac{x+y}{2} \\ 20x + 2y &=x+y \\ 19x + y &= 0 \end{align}$$Clearly this yields no solutions. However. Let's try the number \(x.y\) where the dot represents the decimal place. Then we have:$$ \begin{align} x + \frac{y}{10} &= \frac{x+y}{2} \\10x+y &= 5x+ 5y \\5x-4y &= 0 \end{align}$$This does yield a solution when \(x=4\) and \(y=5\). So the number must be 4.5 which indeed is the average of 4 and 5.

It's just a simple little problem but once you lock into that mindset that the number is an integer a solution seems impossible. I asked Gemini if there are positive numbers that satisfy for the geometric and harmonic means and the short answer is NO. 

27999: The End of a Millennium

A millennium is strictly speaking a period of 1000 years but in my case, dealing as I do in days rather than years, I'm regarding it as a period of 1000 days. Tomorrow I'll be 27999 days old and on Sunday, the 30th of November 2025, I'll turn \( \textbf{28000}\) days old, thus marking a transition from one millennium to another. It must fall on a Sunday of course since the number is divisible by 7, meaning I'll also be 4000 weeks old. 82 days after that I'll celebrate my first palindromic day of the new millennium: 28082.

One of the properties of 28000 is that it is a member of OEIS A085708: Arithmetic derivative of \(10^n\) where \(n=4\). Here is Gemini's account of how the arithmetic derivative is determined.

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

Python

def arithmetic_derivative(n):
    if n < 0: return -arithmetic_derivative(-n)
    if n == 0 or n == 1: return 0
    
    # Simple trial division for factorization
    i = 2
    factors = []
    d = n
    while i * i <= d:
        if d % i:
            i += 1
        else:
            d //= i
            factors.append(i)
    if d > 1:
        factors.append(d)
        
    # Apply logarithmic derivative formula: n' = n * sum(1/p for p in factors)
    # Actually formula is sum( n/p for p in factors ) since p'=1
    # n' = n * sum( p'/p ) = n * sum( 1/p )
    # Let's use the explicit sum rule: (abc...)' = (abc...)(1/a + 1/b + 1/c ...) *if* they are primes? 
    # Let's stick to the definition: (ab)' = a'b + ab'
    # For n = p1 * p2 * ... * pk
    # n' = n * (p1'/p1 + p2'/p2 + ... + pk'/pk)
    # Since pi are primes, pi' = 1.
    # So n' = n * (1/p1 + 1/p2 + ... + 1/pk)
    # Where the factors list includes multiplicity.
    
    result = 0
    for p in factors:
        result += n // p
    return result

val = 10000
der = arithmetic_derivative(val)
print(f"{val=}, {der=}")

Code output

val=10000, der=28000

To find the arithmetic derivative of $10000$, we use the definition and properties of the arithmetic derivative function $n'$.

Rules of Arithmetic Derivative:

1. Prime Rule: For any prime number $p$, $p' = 1$.

2. Product Rule (Leibniz Rule): For any integers $a$ and $b$, $(ab)' = a'b + ab'$.

From these rules, we can derive the Power Rule for a prime $p$ raised to a power $k$:

$$(p^k)' = k p^{k-1}$$

---

Step-by-Step Calculation

Step 1: Prime Factorization

First, find the prime factorization of $10000$.

$$10000 = 10^4 = (2 \times 5)^4 = 2^4 \times 5^4$$

Step 2: Apply the Product Rule

Let $a = 2^4$ and $b = 5^4$. According to the product rule:

$$(2^4 \times 5^4)' = (2^4)' \times 5^4 + 2^4 \times (5^4)'$$

Step 3: Calculate Derivatives of the Factors

Using the power rule $(p^k)' = k p^{k-1}$:

• For $2^4$:

  $$(2^4)' = 4 \times 2^{4-1} \times (2)' = 4 \times 2^3 \times 1 = 4 \times 8 = 32$$

• For $5^4$:

  $$(5^4)' = 4 \times 5^{4-1} \times (5)' = 4 \times 5^3 \times 1 = 4 \times 125 = 500$$

Step 4: Substitute and Simplify

Substitute these derivatives back into the equation from Step 2:

$$10000' = (32) \times 5^4 + 2^4 \times (500)$$

Calculate the terms:

• First term: $32 \times 625 = 20000$

• Second term: $16 \times 500 = 8000$

Step 5: Final Sum

$$10000' = 20000 + 8000 = 28000$$

Alternative Method (Logarithmic Derivative Formula)

For a number $n = p_1^{e_1} \cdots p_k^{e_k}$, the formula is:$$n' = n \sum_{i=1}^k \frac{e_i}{p_i}$$For $10000 = 2^4 \times 5^4$:

$$10000' = 10000 \left( \frac{4}{2} + \frac{4}{5} \right)$$

$$10000' = 10000 \left( 2 + 0.8 \right) = 10000(2.8) = 28000$$

Answer:

The arithmetic derivative of 10000 is 28,000.

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

Courtesy of Gemini, I also learned that the number shows up in the orbital speed of 28,000 km/h that is the specific speed required to maintain a Low Earth Orbit (LEO). 

• This is the cruising speed of the International Space Station (ISS). At this speed, the ISS circles the Earth once every 90 minutes.

• If it moved much slower, gravity would pull it down; much faster, and it would break orbit into deep space.

More disconcertingly, and again thanks to Gemini, the number shows up as the "28,000 Days" Concept in which the number is often used in philosophy and motivational literature to represent the average human lifespan.

• $\dfrac{28,000 \text{ days}}{365 \text{ days}} \approx 76.7 \text{ years}$.

• It is frequently used to visualize the finiteness of life e.g. "You only get 28,000 days".

Let's not forget that the two key digits in 28000 are the 2 and the 8, with the 0's acting as magnifiers of a sort. Thus the focus falls on 28 which of course is a perfect number, being equal to the sum of its proper divisors:$$ \begin{align} 28 &= 1 + 2 + 4 + 7 + 14 \\ 28000 &= 1000 + 2000 + 4000 + 7000 + 14000 \end{align} $$ \( \textbf{Here's what my SageMath algorithm generated:}\)

\(28000 = 2^5 \times 5^3 \times 7\)

There are 48 divisors

The divisors are [1, 2, 4, 5, 7, 8, 10, 14, 16, 20, 25, 28, 32, 35, 40, 50, 56, 70, 80, 100, 112, 125, 140, 160, 175, 200, 224, 250, 280, 350, 400, 500, 560, 700, 800, 875, 1000, 1120, 1400, 1750, 2000, 2800, 3500, 4000, 5600, 7000, 14000, 28000]

The sum of the divisors is \(78624 = 2^5 \times 3^3 \times 7 \times 13\)

The sum of the proper divisors is \(50624 = 2^6 \times 7 \times 113\)

28000 is abundant because its sum of proper divisors is greater than the number itself

Distinct prime factors are [2, 5, 7]

Sum of DISTINCT prime factors of 28000 is \(14 = 2 \times 7\)

28000 is NOT unprimeable because 28001 is prime

Unitary Divisors are [1, 7, 32, 125, 224, 875, 4000, 28000]

28000 is pseudoperfect meaning it is equal to the sum of a PROPER subset of its PROPER divisors

28000 is a Zumkeller number meaning that its divisors can be split into two disjoint sets with same sum

28000 is a gapful number

28000 is a practical number meaning that all smaller numbers can be written as sums of its divisors

The totient is \(9600 = 2^7 \times 3 \times 5^2\)

The cototient is \(28000 - 9600 = 18400 = 2^5 \times 5^2 \times 23\)

The sum of squares of the digits is \(68 = 2^2 \times 17\)

The sum of cubes of the digits is \(520 = 2^3 \times 5 \times 13\)

28000 is a plaindrome in base 6 : 333344

28000 is a nialpdrome in base 8 : 66540

28000 is a xenodrome in base 9 : 42361

Gray Code Equivalent:

\(28000 = 110110101100000 \rightarrow 101101111010000 = 23504 = 2^4 \times 13 \times 113\)

Difference between 28000 and its Gray Code is \(4496 = 2^4 \times 281\)

Binary complement of 28000 is \(4767 = 3 \times 7 \times 227\)

Difference between binary complement and 28000 is \(23233 = 7 \times 3319\)

No energetic or d-powerful classification for 28000.

number is an energetic number

Sum of Squares of the following tuples (if any):

28000 is the sum of the cubes of 10 and 30

Sum of digits is 10 and product of digits is 0

28000 is a Harshad number since it is divisible by its sum of digits 10

\(28000 + 10 = 28010 = 2 \times 5 \times 2801\)

\(28000 - 10  = 27990 = 2 \times 3^2 \times 5 \times 311 \)

\(28000 + 0 = 28000 = 2^5 \times 5^3 \times 7\)

\(28000 - 0 = 28000 = 2^5 \times 5^3 \times 7\)

number + SOD - POD sequence with zero counted is:

[28000, 28010, 28021, 28034, 28051, 28067, 28090, 28109, 28129, 27863, 25873, 24218, 24107, 24121, 24115, 24088, 24110, 24118, 24070, 24083, 24100, 24107]

number + SOD - POD sequence with zero NOT counted is:

[28000, 28010, 28005, 28020, 28000]

The Collatz trajectory requires 33 steps required to reach 1 with trajectory:

[28000, 14000, 7000, 3500, 1750, 875, 2626, 1313, 3940, 1970, 985, 2956, 1478, 739, 2218, 1109, 3328, 1664, 832, 416, 208, 104, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1]

The Aliquot Sequence for 28000 requires 79 steps and is:

[28000, 50624, 65200, 92404, 81840, 203856, 343728, 894288, 1494448, 1648208, 1649200, 3271120, 4585520, 6681616, 7404784, 7405776, 17989424, 17990416, 22007024, 25406608, 25867888, 25868880, 70077360, 154194000, 359582640, 859196496, 1455083952, 2979489360, 7116150192, 11860254288, 22402715440, 40354410704, 40354411696, 40737128272, 40819290362, 23229175558, 11614587782, 10147429498, 5602405262, 2801714530, 2259985694, 1216520482, 651851870, 521481514, 260870486, 144688954, 72344480, 111721864, 97756646, 60157978, 30189062, 17477938, 10281194, 9165718, 4582862, 2362594, 1500566, 848218, 640742, 325258, 162632, 153268, 114958, 58922, 34714, 20474, 11386, 5696, 5734, 3194, 1600, 2337, 1023, 513, 287, 49, 8, 7, 1, 0]

The Anti-Divisors of 28000 are:

[3, 11, 29, 33, 64, 320, 448, 1600, 1697, 1931, 2240, 5091, 8000, 11200, 18667]

The Arithmetic Derivative of 28000 is 90800

Difference between Arithmetic Derivative and 28000 is \(62800 = 2^4 \times 5^2 \times 157\)

The Maximum - Minimum Recursive Algorithm for 28000 produces:

[28000, 81972, 85932, 74943, 62964, 71973, 83952, 74943]

The Minimal Goldbach decomposition of 28000 is 3 and 27997

28000 requires 1 steps to reach the palindrome 28082 under the Reverse and Add algorithm

The number of steps required is to reach home prime is 3 :

[28000, 222225557, 293758449, 31931166247]

The additive digital root of 28000 is 1 and is derived as follows:

[28000, 10, 1]

The multiplicative digital root of 28000 is 0 and derived as follows:

[28000, 0]

The steps on the Descent to Zero are:

[28000, 16000, 6000, 0]

The number of steps required is 3

The multiplicative persistence of 28000 is as follows (NOT taking zeros into account):

[28000, 16, 6]

Trajectory of 28000 under ODD(+) and EVEN(-) algorithm is:

[28000, 27990, 28013, 28007, 28004, 27990]

28000 is a captive of the vortex beginning and ending with 27990

Trajectory of 28000 under EVEN(+) and ODD(-) algorithm is:

[28000, 28010, 28019, 28019]

28000 is a captive of the attractor 28019

Trajectory under the Primes(+) and Non_Primes(-) trajectory is:

[28000, 27994, 27981, 27972, 27981]

28000 is a captive of the vortex beginning and ending with 27981

Determinant is -60 for Circulant Matrix:

Difference between Determinant and 28000 is \(28060 = 2^2 \times 5 \times 23 \times 61\)

[ 2     8     0     0     0 ] 

[ 8     0     0     0     2 ]

[ 0     0     0     2     8 ]

[ 0     0     2     8     0 ]

[ 0     2     8     0     0 ]

Here is how the number holds up under \( \textbf{Conway's Game of Life} \):

Here's what Numbers Aplenty had to say about the number: