An Interesting Intersection

I've never encountered an intersection of the arithmetic derivative and the totient function before until looking at one of the properties associated with the number 27620, my diurnal age yesterday. This number has an interesting property that qualifies it for membership in OEIS A352332:

A352332: numbers \(k\) for which \(k = \phi(k') + \phi(k'') \), where \( \phi \) is the Euler totient function (A000010), \(k'\) is the arithmetic derivative of \(k\) (A003415) and \(k''\) is the second arithmetic derivative of \(k\) (A068346).

Let's look at 27620 and its arithmetic derivatives (denoted by ' and '') and their totients (denoted by \( \phi \) ). Firstly:$$ \begin{align} 27620 &= 2^2 \times 5 \times 1381\\  (27620)' &=33164 \\(27620)'' &= (33164)'\\ &=33168 \\ \phi(33164) &= 16580 \\ \phi(33168) &= 11040 \\ 27620 &= 16580 + 11040 \end{align} $$Up to 40000, the members of this sequence are 4, 260, 294, 740, 1460, 3140, 3860, 5540, 8420, 10820, 15140, 19940, 21860, 24020, 24260, 27620 and 37460 (permalink). Clearly, numbers satisfying these criteria are few and far between.

This got me thinking about whether there were any numbers that satisfied the criterion that they were simply the sum of their first and second arithmetic derivatives, ignoring the totient function. There are only three numbers in the range up to 100,000 that satisfy. They are 6, 42, 15590 and 47058. Here is the breakdown:$$ \begin{align} 6 &= 5 + 1\\42 &= 41 + 1\\15590 &= 10923 + 466\\47058 &= 47057 + 1 \end{align}$$It can be noted that in all but 15590, the first derivative yields a prime and thus the second derivative is 1.

What if, instead of the totient, we use the sum of the divisors (\(\sigma\)) of the number? In the range up to 100,000, no numbers satisfy but if we use the sum of the PROPER divisors, then a single number qualifies and that is 50 (permalink). Let's look more closely:$$ \begin{align} 50 &= 2 \times 5^2 \\ (50)' &= 45 \\ \sigma(45) -45&= 78 - 45 \\ &=33\\(50)'' &=(45)'\\ &=39\\ \sigma(39)-39 &= 56-39\\ &= 17 \\50 &= 33 +17 \end{align} $$It should be noted that numbers like 50, 15590 and 27620 with the specific properties that have been mentioned in this post retain these properties in bases other than 10. In other words, the properties are base independent.