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 numbers related to a given number and I've included a list of such numbers 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