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.