The arithmetic derivative is a derivation on $\mathbb{Z}$ that is $1$ for all prime numbers. On positive integers other than 1, this always returns a positive integer. My question is, is there a way to interpret the arithmetic derivative as counting something?
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3OEIS says that: "Let n be the product of a multiset P of k primes. Consider the k-dimensional box whose edges are the elements of P. Then the (k-1)-dimensional surface of this box is 2a(n). For example, 2a(25) = 20, the perimeter of a 5 X 5 square. Similarly, 2a(18) = 42, the surface area of a 2 X 3 X 3 box. - David W. Wilson, Mar 11 2011" – Phicar Jun 28 '23 at 20:48
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https://oeis.org/A003415 – Dan Jun 28 '23 at 20:53