Sums of Digits

The number associated with my diurnal age today, 28248, has the property that its digits raised to the fourth power are equal to its totient.$$ \begin{align} 2^4 + 8^4 + 2^4 + 4^4 + 8^4 &= 8480 \\ \phi(28248) &= 8480 \end{align}$$This qualifies it for membership in OEIS A269669:

A269669 numbers whose Euler totient function is equal to the sum of some fixed power of their digits.

The numbers in this sequence are few and far between, with 28248 being the last in the range up to 40000. The initial members are 1, 2, 20, 40, 228, 352, 712, 813, 835, 2079, 4020, 28248. Take 20 as another example:$$ \begin{align} 2^3 + 0^3 &= 8 \\ \phi(20) &= 8 \end{align}$$As part of my daily number analysis, I consider:

• the sum of the digits of a number
• the sum of the digits squared of a number
• the sum of the digits cubed of a number

If all three sums are prime, then I make a note of it. I consider higher powers of the digits in the context of narcissistic numbers where a narcissistic number is defined as a \(k\)-digit nonnegative number equal to the sum of the \(k\)-th powers of its digits. An example is of such a number is 9474 where:$$9474=9^4+4^4+7^4+4^4$$D-powerful numbers are similar but are defined as integers that can be expressed as a sum of positive powers of their digits. An example is 994 where:$$994=9^3+9^1+4^4$$OEIS A269669 relates such sums to a number's totient which I hadn't thought of doing before. This idea can be extended to other number properties of a number and I've included a list of such properties as output from my daily number analysis. The example below is for 28248 where the equality between its totient and the sum of its digits raised to the fourth power can be clearly seen.

  Daily Number                      28248
  Sum of Divisors                   77760
  Sum of Proper Divisors            49512
  Totient                           8480
  Sum of DISTINCT prime factors     123
  Sum of Unitary Divisors           46656
  Sum of Digits                     24
  Sum of Digits Squared             152
  Sum of Digits Cubed               1104
  Sum of Digits to Fourth Power     8480
  Sum of Digits to Fifth Power      66624
  Product of Digits                 1024
  Gray Code                         22900
  Binary Complement                 4519
  Arithmetic Derivative             54620
  Determinant of Circulant Matrix   11904

If we search for numbers whose arithmetic derivative is equal to the sum of some fixed power of their digits, we don't find many. In the case of \(n=2\), we only have 581, 8549 and 16999 with sums of digits squared and arithmetic derivatives of 90,186 and 280 respectively in the range up to 40000. For \(n=3\) we only have 142, 6127 and 12643 with sums of digits cubed and arithmetic derivatives of 73, 568 and 316 respectively. For higher powers, nothing in the range up to 40000. Here is the permalink.$$ \begin{align} 581 &\rightarrow 5^2+8^2+1^2 = 90 \text{ = arithmetic derivative of 581} \\ 8549 &\rightarrow 8^2+5^2+4^2+9^2 = 186 \text{ = arithmetic derivative of 8549} \\ 16999 &\rightarrow 1^2+6^2+9^2+9^2+9^2 = 280 \text{ = arithmetic derivative of 16999} \\ 142 &\rightarrow 1^2+4^3+2^3 = 73 \text{ = arithmetic derivative of 142} \\ 6127 &\rightarrow 6^3+1^3+2^3+7^3 = 568 \text{ = arithmetic derivative of 6127} \\ 12643 &\rightarrow 1^3+2^3+6^3+4^3+3^3 =316 \text{ = arithmetic derivative of 12643} \end{align}$$

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.

Perfect Powers

Firstly a reminder. What is a perfect power?

A perfect power is a positive integer that can be expressed as an integer power of another positive integer. More formally, \(n\) is a perfect power if there exist natural numbers \(m > 1\), and \(k > 1\) such that \(m^k = n\). In this case, \(n\) may be called a perfect \(k\)-th power. If \(k = 2\) or \(k = 3\), then \(n\) is called a perfect square or perfect cube, respectively. Sometimes 1 is also considered a perfect power (\(1^k = 1\) for any \(k\)).

Today I turned 28188 days old and one of the interesting properties of this number, apart from the digit 8 appearing three times, is that its sum of digits (SOD) and product of digits (POD) are both perfect powers. This is because:$$ \begin{align} \text{SOD}(28188) &= 27 = 3^3\\ \text{POD}(28188) &= 1024 = 2^{10 }\end{align}$$This got me thinking as to how many positive integers in the range up to 40000 have this property. Well, it turns out that there are 217 and here they are (permalink):

4, 8, 9, 18, 22, 44, 81, 88, 144, 224, 242, 333, 414, 422, 441, 448, 484, 844, 999, 1124, 1133, 1142, 1177, 1214, 1224, 1241, 1242, 1313, 1331, 1339, 1393, 1412, 1421, 1422, 1555, 1717, 1771, 1888, 1933, 2114, 2124, 2141, 2142, 2214, 2222, 2241, 2248, 2284, 2411, 2412, 2421, 2428, 2482, 2824, 2842, 3113, 3131, 3139, 3193, 3311, 3319, 3391, 3913, 3931, 4112, 4121, 4122, 4211, 4212, 4221, 4228, 4282, 4444, 4822, 5155, 5515, 5551, 7117, 7171, 7711, 8188, 8224, 8242, 8422, 8818, 8881, 8888, 9133, 9313, 9331, 11114, 11124, 11133, 11141, 11142, 11214, 11222, 11241, 11248, 11284, 11313, 11331, 11411, 11412, 11421, 11428, 11482, 11824, 11842, 12114, 12122, 12141, 12148, 12184, 12212, 12221, 12222, 12411, 12418, 12481, 12814, 12841, 12888, 13113, 13131, 13311, 13399, 13939, 13993, 14111, 14112, 14121, 14128, 14182, 14211, 14218, 14281, 14488, 14812, 14821, 14848, 14884, 18124, 18142, 18214, 18241, 18288, 18412, 18421, 18448, 18484, 18828, 18844, 18882, 19339, 19393, 19933, 21114, 21122, 21141, 21148, 21184, 21212, 21221, 21222, 21411, 21418, 21481, 21814, 21841, 21888, 22112, 22121, 22122, 22211, 22212, 22221, 22228, 22282, 22444, 22822, 24111, 24118, 24181, 24244, 24424, 24442, 24811, 28114, 28141, 28188, 28222, 28411, 28818, 28881, 31113, 31131, 31311, 31399, 31939, 31993, 33111, 33199, 33399, 33919, 33939, 33991, 33993, 39139, 39193, 39319, 39339, 39391, 39393, 39913, 39931, 39933

What about numbers whose sums of divisors are perfect powers? In the range up to 40000, there are only 18 and they are (permalink):

3, 7, 21, 31, 81, 93, 127, 217, 381, 400, 651, 889, 2667, 3937, 8191, 11811, 24573, 27559

Take 21 with divisors of 1, 3, 7 and 21 as an example:$$ \sigma(21)=32=2^5$$What about numbers whose totients are perfect powers? Let's recall that:

The totient of a number, denoted by the Euler's totient function \(\phi(n)\), is the count of positive integers less than or equal to \(n\) that are coprime (relatively prime) to \(n\). Two numbers are coprime if their only common factor is 1.

Numbers of this sort are more frequent with there being 119 in the range up to 40000. They are (permalink):

5, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, 272, 320, 340, 384, 408, 480, 510, 512, 514, 544, 640, 680, 768, 771, 816, 960, 1020, 1024, 1028, 1088, 1280, 1285, 1360, 1536, 1542, 1632, 1920, 2040, 2048, 2056, 2176, 2560, 2570, 2720, 3072, 3084, 3264, 3840, 3855, 4080, 4096, 4112, 4352, 4369, 5120, 5140, 5440, 6144, 6168, 6528, 7680, 7710, 8160, 8192, 8224, 8704, 8738, 10240, 10280, 10880, 12288, 12336, 13056, 13107, 15360, 15420, 16320, 16384, 16448, 17408, 17476, 20480, 20560, 21760, 21845, 24576, 24672, 26112, 26214, 30720, 30840, 32640, 32768, 32896, 34816, 34952

Let's take 20 as an example. The numbers that are coprime to 20 are 1, 3, 7, 9, 11, 13, 17, 19 and so we have we have a count of 8:$$ \phi(20)=8=2^3$$Next, let's consider the sum of the anti-divisors of a number. First let's recall what an antidivisor is using Gemini to explain in very clear terms:

An anti-divisor is a number that fails to divide a target integer by the largest possible margin. While a regular divisor divides a number evenly with no remainder, an anti-divisor leaves the most unbiased, centered remainder possible. [1, 2, 3]

The concept was first formalized by mathematician Jon Perry. [1]

Mathematical Rules

An integer \(d\) (where \(1 < d < n\)) is an anti-divisor of \(n\) if it satisfies one of the following rules: [1, 2]

• For even values of \(d\): The number \(d\) divides \(2n\) evenly. Mathematically, we have:$$n \bmod d = \dfrac{d}{2}$$
• For odd values of \(d\): The number \(d\) divides \(2n - 1\) or \(2n + 1\) evenly. Mathematically we have  [1, 2]:$$n \bmod d = \dfrac{d-1}{2}  \text{ or } n \bmod d = \dfrac{d+1}{2}$$

Visual Example: Anti-divisors of 20

To find the anti-divisors of \(20\), you look for numbers between \(2\) and \(19\) that fail to divide \(20\), but cleanly divide \(40\) (twice \(20\)) or \(39\)/\(41\). [1]

• 3 is an odd anti-divisor because \(3 \times 13 = 39\), leaving a gap of \(1\) (since \(\frac{3-1}{2} = 1\), and \(20 \bmod 3 = 2\)).
• 8 is an even anti-divisor because \(8 \times 5 = 40\). It divides \(2 \times 20\) perfectly.
• 13 is an odd anti-divisor because \(13 \times 3 = 39\). [1]

Key Facts

• The number 1 is never an anti-divisor, as it evenly divides all integers.
• The number 2 has no anti-divisors.
• Prime numbers have a limited number of anti-divisors based heavily on multiples of \(2n\). [1, 2, 3]

For a complete look at integer sequences and properties, you can explore the The Anti-Divisor - OEIS catalog. [1]

There are 20 numbers in the range up to 40000 that satisfy and these are (permalink):

6, 8, 9, 14, 36, 89, 96, 221, 541, 576, 740, 778, 1854, 2114, 2571, 10277, 13631, 16160, 16389, 39428

Let's take 39428 as an example. The antidivisors of this number are:

3, 5, 7, 8, 15, 21, 35, 105, 751, 2253, 3755, 5257, 11265, 15771, 26285

The sum of these anti-divisors is \(65536 = 2^{16}\) and so it is perfect power.

Further investigation could target:

• the sum of the proper divisors
• the sum of the non-divisors
• the determinant of the circulant matrix
• the arithmetic derivative

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

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:$$0 \rightarrow 28124 \text{ requires } 34183 \text{ (prime) iterations}$$

Prime Factor Fibonacci

A recent number, 27960, associated with my diurnal age has an interesting property if you look at the prime factors more closely:$$27960=2^3 \times 3 \times 5 \times 233$$Let's ignore multiplicity and look at the sum of digits of each distinct prime factor. The 2, 3 and 5 remain the same but 233 becomes 8 and the sequence of prime factors then becomes:$$2, 3, 5, 8$$This is the Fibonacci sequence. How many numbers with four prime factors (ignoring multiplicity) have this property. Well, in the range up to 40000, it turns out that there are 115. These numbers are (permalink):

510, 1020, 1530, 1590, 2040, 2130, 2550, 3060, 3180, 3210, 4080, 4260, 4590, 4770, 5100, 6120, 6360, 6390, 6420, 6990, 7314, 7530, 7650, 7950, 8160, 8520, 8670, 9180, 9540, 9630, 9798, 10200, 10650, 12240, 12720, 12750, 12780, 12840, 12930, 13038, 13770, 13980, 14310, 14628, 14766, 15060, 15090, 15300, 15630, 15900, 16050, 16320, 17040, 17085, 17340, 17466, 18360, 19080, 19170, 19260, 19596, 20400, 20970, 21030, 21300, 21942, 22590, 22950, 23850, 24480, 25440, 25500, 25560, 25680, 25860, 26010, 26076, 26322, 27540, 27960, 28620, 28890, 29256, 29394, 29532, 30120, 30180, 30600, 31260, 31800, 31830, 31950, 32100, 32154, 32637, 32640, 34080, 34530, 34638, 34680, 34932, 34950, 35445, 35511, 36690, 36720, 37650, 38160, 38250, 38340, 38520, 38790, 39114, 39192, 39750

Lets consider the last number in this list: 39750. In this case we have:$$ \begin{align} 39750 &= 2 \times 3 \times 5^3 \times 53 \\ & \rightarrow 2, 3, 5, 8 \end{align}$$If we extend the number of prime factors to five, then only one number satisfies in the range up to 40000:$$ \begin{align}34170 &= 2 \times 3 \times 5 \times 17 \times 67 \\ &\rightarrow 2, 3, 5, 8, 13 \end{align}$$However, there are 160 numbers that satisfy in the range up to one million (permalink). The algorithm actually looks for generalised Fibonacci sequences:$$a,b,c,d,e, \dots \text{ such that } c = a+b, d=b+c, e=c+d \dots$$However, all of the 160 numbers begin with 2. For example, the last of the numbers is 9988520:$$ \begin{align} 998520 &= 2^3 \times 3 \times 5 \times 53 \times 157 \\ &\rightarrow 2, 3, 5, 8, 13 \end{align} $$I tried with six prime factors but without success up to 100 million. Finally I realised that any suitable number must add to 21 and thus be divisible by 3. Such a number can never be a prime factor and so the prime factor sequence must end at 13 and can never progress further.

Sequences Involving SOD and POD

The number associated with my diurnal age today, 27923, has the interesting property that its sum of digits (23) is equal to the last two digits of the number. This number is part of a sequence of consecutive numbers that all share this same property. The numbers are 27920 up to 27929. In the range of numbers up to 40000, there are 440 numbers with this property. They are (permalink):

SOD = Concatenation of Last Two Digits

910, 911, 912, 913, 914, 915, 916, 917, 918, 919, 1810, 1811, 1812, 1813, 1814, 1815, 1816, 1817, 1818, 1819, 2710, 2711, 2712, 2713, 2714, 2715, 2716, 2717, 2718, 2719, 3610, 3611, 3612, 3613, 3614, 3615, 3616, 3617, 3618, 3619, 4510, 4511, 4512, 4513, 4514, 4515, 4516, 4517, 4518, 4519, 5410, 5411, 5412, 5413, 5414, 5415, 5416, 5417, 5418, 5419, 6310, 6311, 6312, 6313, 6314, 6315, 6316, 6317, 6318, 6319, 7210, 7211, 7212, 7213, 7214, 7215, 7216, 7217, 7218, 7219, 8110, 8111, 8112, 8113, 8114, 8115, 8116, 8117, 8118, 8119, 9010, 9011, 9012, 9013, 9014, 9015, 9016, 9017, 9018, 9019, 9920, 9921, 9922, 9923, 9924, 9925, 9926, 9927, 9928, 9929, 10810, 10811, 10812, 10813, 10814, 10815, 10816, 10817, 10818, 10819, 11710, 11711, 11712, 11713, 11714, 11715, 11716, 11717, 11718, 11719, 12610, 12611, 12612, 12613, 12614, 12615, 12616, 12617, 12618, 12619, 13510, 13511, 13512, 13513, 13514, 13515, 13516, 13517, 13518, 13519, 14410, 14411, 14412, 14413, 14414, 14415, 14416, 14417, 14418, 14419, 15310, 15311, 15312, 15313, 15314, 15315, 15316, 15317, 15318, 15319, 16210, 16211, 16212, 16213, 16214, 16215, 16216, 16217, 16218, 16219, 17110, 17111, 17112, 17113, 17114, 17115, 17116, 17117, 17118, 17119, 18010, 18011, 18012, 18013, 18014, 18015, 18016, 18017, 18018, 18019, 18920, 18921, 18922, 18923, 18924, 18925, 18926, 18927, 18928, 18929, 19820, 19821, 19822, 19823, 19824, 19825, 19826, 19827, 19828, 19829, 20710, 20711, 20712, 20713, 20714, 20715, 20716, 20717, 20718, 20719, 21610, 21611, 21612, 21613, 21614, 21615, 21616, 21617, 21618, 21619, 22510, 22511, 22512, 22513, 22514, 22515, 22516, 22517, 22518, 22519, 23410, 23411, 23412, 23413, 23414, 23415, 23416, 23417, 23418, 23419, 24310, 24311, 24312, 24313, 24314, 24315, 24316, 24317, 24318, 24319, 25210, 25211, 25212, 25213, 25214, 25215, 25216, 25217, 25218, 25219, 26110, 26111, 26112, 26113, 26114, 26115, 26116, 26117, 26118, 26119, 27010, 27011, 27012, 27013, 27014, 27015, 27016, 27017, 27018, 27019, 27920, 27921, 27922, 27923, 27924, 27925, 27926, 27927, 27928, 27929, 28820, 28821, 28822, 28823, 28824, 28825, 28826, 28827, 28828, 28829, 29720, 29721, 29722, 29723, 29724, 29725, 29726, 29727, 29728, 29729, 30610, 30611, 30612, 30613, 30614, 30615, 30616, 30617, 30618, 30619, 31510, 31511, 31512, 31513, 31514, 31515, 31516, 31517, 31518, 31519, 32410, 32411, 32412, 32413, 32414, 32415, 32416, 32417, 32418, 32419, 33310, 33311, 33312, 33313, 33314, 33315, 33316, 33317, 33318, 33319, 34210, 34211, 34212, 34213, 34214, 34215, 34216, 34217, 34218, 34219, 35110, 35111, 35112, 35113, 35114, 35115, 35116, 35117, 35118, 35119, 36010, 36011, 36012, 36013, 36014, 36015, 36016, 36017, 36018, 36019, 36920, 36921, 36922, 36923, 36924, 36925, 36926, 36927, 36928, 36929, 37820, 37821, 37822, 37823, 37824, 37825, 37826, 37827, 37828, 37829, 38720, 38721, 38722, 38723, 38724, 38725, 38726, 38727, 38728, 38729, 39620, 39621, 39622, 39623, 39624, 39625, 39626, 39627, 39628, 39629

A plot of these numbers shows a quite regular pattern. See Figure 1.

Figure 1: permalink

Addendum: I noticed that OEIS A052018: numbers \(k\) with the property that the sum of the digits of \(k\) is a substring of \(k\) is a more general version of what I've just described.

If we consider the product of digits instead and exclude numbers with the digit 0, then only 31 numbers satisfy in the range up to 40000. These are (permalink):

POD = Concatenation of Last Two Digits

236, 315, 324, 612, 1236, 1315, 1324, 1612, 2136, 2312, 3115, 3124, 3212, 6112, 11236, 11315, 11324, 11612, 12136, 12312, 13115, 13124, 13212, 16112, 21136, 21312, 23112, 31115, 31124, 31212, 32112

Figure 2 shows a graph of these numbers.

Figure 2: permalink

Primes from SOD and POD

I'm using the acronym SOD to stand for Sum Of Digits and POD to stand for Product Of Digits. In a blog June 2021 post titled SOD ET AL, I made mention of OEIS A047791:

A047791

Numbers  such that  plus digit sum of  (A007953) equals a prime.

In March of 2024, I made a blog post titled More Sequences Involving SOD and POD in which I looked at semiprimes and sphenic numbers that remain semiprimes and sphenic numbers when their sums of digits and products of digits are added to them. For some reason, I've never looked at numbers that become prime when their sum of digits is added to them and that also become prime when their product of digits is added to them. When considering the product of digits of a number, it's usual to exclude numbers containing a zero because then the product of digits is alway zero. However, the decision can be made to exclude any zero digits in the multiplication.

Let's consider the first approach where numbers containing a zero are excluded from consideration. How many numbers \(n\) in the range up to 40000, satisfy the following criteria:

• \(n\) does not contain the digit 0
• \(n\) + SOD(\(n )\) \( \rightarrow \) a prime number
• \(n\) + POD(\(n )\) \( \rightarrow \) a prime number

There are 351 such numbers (permalink):

1, 163, 233, 253, 293, 341, 343, 431, 473, 493, 499, 563, 611, 617, 743, 767, 923, 1241, 1421, 1423, 1483, 1489, 1849, 1867, 2147, 2231, 2233, 2273, 2327, 2369, 2413, 2543, 2569, 2633, 2639, 2783, 2839, 3287, 3449, 3649, 3661, 3829, 4121, 4211, 4217, 4273, 4387, 4459, 4529, 4547, 4729, 4921, 5263, 5461, 5627, 5663, 5861, 6121, 6127, 6181, 6527, 6529, 6617, 6637, 6653, 6743, 6761, 6857, 6949, 7223, 7429, 7681, 8273, 8431, 8491, 8569, 8671, 8789, 8837, 8839, 8899, 8983, 9263, 9641, 9649, 9689, 9869, 9881, 9889, 11233, 11237, 11251, 11297, 11341, 11567, 11581, 11611, 11657, 11677, 11897, 11899, 12151, 12379, 12443, 12553, 12557, 12667, 12797, 12977, 13163, 13211, 13231, 13321, 13453, 13457, 13523, 13547, 13673, 13697, 13729, 13837, 13877, 13879, 13891, 13969, 14137, 14311, 14353, 14423, 14429, 14467, 14483, 14533, 14537, 14623, 14641, 14647, 14689, 14957, 15121, 15167, 15217, 15257, 15277, 15361, 15413, 15451, 15491, 15619, 15721, 15727, 15769, 15781, 15859, 15947, 16319, 16391, 16427, 16513, 16577, 16799, 16993, 16997, 17183, 17299, 17329, 17837, 17879, 18197, 18229, 18287, 18517, 18559, 18751, 19381, 19411, 19457, 19523, 19549, 19583, 19673, 19691, 19741, 19831, 19859, 21311, 21379, 21467, 21511, 21577, 21593, 21737, 21751, 21977, 21991, 22121, 22123, 22259, 22369, 22387, 22549, 22657, 22747, 22837, 22921, 22981, 23117, 23179, 23269, 23353, 23519, 23573, 23599, 23719, 23731, 23791, 24259, 24343, 24499, 24527, 24761, 24949, 25111, 25153, 25223, 25283, 25319, 25333, 25441, 25517, 25519, 25577, 25681, 25771, 25847, 25913, 25991, 25997, 26233, 26251, 26273, 26323, 26611, 26699, 26927, 27157, 27263, 27317, 27427, 27511, 27887, 27931, 28181, 28213, 28853, 28877, 28981, 29113, 29117, 29153, 29171, 29179, 29281, 29357, 29597, 29621, 29731, 29887, 29933, 31141, 31213, 31217, 31493, 31613, 31651, 31697, 31789, 31837, 31853, 31879, 32111, 32173, 32281, 32357, 32447, 32539, 32687, 32689, 32957, 32971, 33383, 33413, 33527, 33547, 33581, 33587, 33769, 33851, 34247, 34313, 34339, 34463, 34577, 34667, 34681, 34793, 34861, 35143, 35251, 35257, 35323, 35417, 35521, 35569, 35699, 35783, 35831, 35873, 35981, 36229, 36287, 36469, 36559, 36661, 36919, 36953, 36991, 37321, 37369, 37547, 37613, 37637, 37871, 38221, 38351, 38683, 38689, 38719, 38887, 38939, 38951, 38959, 39161, 39361, 39493, 39521, 39587, 39653, 39691, 39943, 39947

The number associated with my diurnal age today, 27887, can be found in this list because:

• SOD(27887) = 32 and POD(27887) = 6272
• 27887 + 32 = 27919 which is a prime number
• 27887 + 6272 = 34159 which is a prime number

If we require that the initial number \(n\) be prime, then only 136 numbers qualify. These numbers thus meet the following criteria:

• \(n\) is prime and does not contain the digit 0
• \(n\) + SOD(\(n) \) \( \rightarrow \) a prime number 
• \(n\) + POD(\(n) \) \( \rightarrow \) a prime number 

Here are the numbers (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

If we decide to include numbers containing the digit 0 (but exclude them in the multiplication) then 444 numbers qualify in the range up to 40000. For example, 10 qualifies since SOD(10) = 1 and POD(101) = 1 and 11 is prime. The criteria to be satisfied are thus:

• \(n\) + SOD(\(n) \) \( \rightarrow \) a prime number 
• \(n\) + POD(\(n) \) \( \rightarrow \) a prime number with 0 excluded from multiplication

Here are the numbers (permalink):

1, 10, 100, 163, 233, 253, 293, 341, 343, 431, 473, 493, 499, 563, 611, 617, 743, 767, 923, 1241, 1421, 1423, 1483, 1489, 1601, 1603, 1849, 1867, 2053, 2147, 2231, 2233, 2273, 2327, 2369, 2413, 2543, 2569, 2633, 2639, 2783, 2839, 2903, 3170, 3190, 3287, 3449, 3607, 3649, 3661, 3829, 3970, 4121, 4211, 4217, 4273, 4387, 4459, 4529, 4547, 4729, 4903, 4909, 4921, 5063, 5263, 5461, 5627, 5663, 5861, 6037, 6053, 6059, 6073, 6103, 6121, 6127, 6181, 6509, 6527, 6529, 6617, 6637, 6653, 6703, 6743, 6761, 6857, 6949, 7043, 7190, 7223, 7310, 7403, 7429, 7681, 8059, 8273, 8431, 8491, 8569, 8671, 8789, 8837, 8839, 8899, 8983, 9043, 9263, 9310, 9403, 9409, 9641, 9649, 9689, 9869, 9881, 9889, 10061, 10243, 10261, 10421, 10447, 10669, 10843, 10847, 11233, 11237, 11251, 11297, 11341, 11567, 11581, 11611, 11657, 11677, 11897, 11899, 12041, 12061, 12151, 12379, 12401, 12443, 12553, 12557, 12601, 12667, 12797, 12809, 12977, 13163, 13211, 13231, 13321, 13453, 13457, 13523, 13547, 13673, 13697, 13700, 13729, 13837, 13877, 13879, 13891, 13969, 14021, 14041, 14137, 14311, 14353, 14407, 14423, 14429, 14467, 14483, 14533, 14537, 14623, 14641, 14647, 14689, 14807, 14957, 15121, 15167, 15217, 15257, 15277, 15361, 15413, 15451, 15491, 15619, 15721, 15727, 15769, 15781, 15859, 15947, 16067, 16319, 16391, 16427, 16513, 16577, 16799, 16993, 16997, 17183, 17299, 17329, 17837, 17879, 18197, 18203, 18209, 18229, 18287, 18517, 18559, 18751, 19381, 19411, 19457, 19523, 19549, 19583, 19673, 19691, 19700, 19741, 19831, 19859, 20093, 20141, 20273, 20323, 20327, 20369, 20491, 20729, 20923, 20927, 21311, 21379, 21467, 21511, 21577, 21593, 21601, 21737, 21751, 21803, 21977, 21991, 22121, 22123, 22259, 22369, 22387, 22549, 22657, 22703, 22747, 22837, 22921, 22981, 23003, 23083, 23117, 23179, 23269, 23353, 23519, 23573, 23599, 23719, 23731, 23791, 24011, 24077, 24101, 24259, 24343, 24499, 24503, 24527, 24761, 24949, 25003, 25043, 25111, 25153, 25223, 25283, 25319, 25333, 25441, 25517, 25519, 25577, 25681, 25771, 25847, 25913, 25991, 25997, 26093, 26101, 26233, 26251, 26273, 26323, 26611, 26699, 26927, 27043, 27157, 27263, 27317, 27407, 27427, 27511, 27887, 27931, 28181, 28213, 28303, 28853, 28877, 28981, 29113, 29117, 29153, 29171, 29179, 29281, 29357, 29597, 29621, 29731, 29887, 29933, 30130, 30449, 30481, 30487, 31070, 31141, 31213, 31217, 31493, 31613, 31651, 31697, 31789, 31837, 31853, 31879, 32111, 32173, 32281, 32357, 32447, 32539, 32687, 32689, 32957, 32971, 33383, 33413, 33527, 33547, 33581, 33587, 33769, 33851, 34247, 34313, 34339, 34409, 34463, 34577, 34609, 34667, 34681, 34793, 34861, 35143, 35251, 35257, 35323, 35417, 35521, 35569, 35699, 35783, 35831, 35873, 35981, 36023, 36047, 36229, 36287, 36469, 36559, 36607, 36661, 36919, 36953, 36991, 37070, 37300, 37321, 37369, 37547, 37613, 37637, 37700, 37871, 38023, 38221, 38351, 38683, 38689, 38719, 38801, 38807, 38887, 38939, 38951, 38959, 39161, 39361, 39493, 39521, 39587, 39653, 39691, 39943, 39947

If the initial number is required to be prime, then only 170 numbers satisfy the following criteria:

• \(n\) is prime
• \(n\) + SOD(\(n) \) \( \rightarrow \) a prime number 
• \(n\) + POD(\(n) \) \( \rightarrow \) a prime number with 0 excluded from multiplication

Here are the numbers (permalink):

163, 233, 293, 431, 499, 563, 617, 743, 1423, 1483, 1489, 1601, 1867, 2053, 2273, 2543, 2633, 2903, 3449, 3607, 4211, 4217, 4273, 4547, 4729, 4903, 4909, 5861, 6037, 6053, 6073, 6121, 6529, 6637, 6653, 6703, 6761, 6857, 6949, 7043, 7681, 8059, 8273, 8431, 8837, 8839, 9043, 9403, 9649, 9689, 10061, 10243, 10847, 11251, 11657, 11677, 11897, 12041, 12379, 12401, 12553, 12601, 12809, 13163, 13457, 13523, 13697, 13729, 13877, 13879, 14407, 14423, 14533, 14537, 14957, 15121, 15217, 15277, 15361, 15413, 15451, 15619, 15727, 15859, 16067, 16319, 16427, 16993, 17183, 17299, 17837, 18229, 18287, 18517, 19381, 19457, 19583, 20323, 20327, 20369, 21379, 21467, 21577, 21601, 21737, 21751, 21803, 21977, 21991, 22123, 22259, 22369, 22549, 22921, 23003, 23117, 23269, 23599, 23719, 24077, 24499, 24527, 25111, 25153, 25577, 25771, 25847, 25913, 25997, 26251, 26699, 26927, 27043, 27407, 27427, 28181, 29153, 29179, 30449, 32173, 32687, 32957, 32971, 33413, 33547, 33581, 33587, 33769, 33851, 34313, 34667, 35251, 35257, 35323, 35521, 35569, 35831, 36229, 36469, 36559, 36607, 36919, 37321, 37369, 37547, 37871, 38351, 38959, 39161, 39521