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.$$ \begin{align} 581 &\rightarrow 5^2+8^2+1^2 = 90 \text{ = arithmetic derivative of 581} \\ 8549 &\rightarrow 8^2+5^2+4^2+9^2 = 186 \text{ = arithmetic derivative of 8549} \\ 16999 &\rightarrow 1^2+6^2+9^2+9^2+9^2 = 280 \text{ = arithmetic derivative of 16999} \\ 142 &\rightarrow 1^2+4^3+2^3 = 73 \text{ = arithmetic derivative of 142} \\ 6127 &\rightarrow 6^3+1^3+2^3+7^3 = 568 \text{ = arithmetic derivative of 6127} \\ 12643 &\rightarrow 1^3+2^3+6^3+4^3+3^3 =316 \text{ = arithmetic derivative of 12643} \end{align}$$
