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For n = 4, A003415(4) = 4, thus as it is among the fixed points, of A003415 and a(4) = 4.

For n = 21 = 3*7, A051903(21) = 1, the first derivative A003415(21) = 10 = 2*5, , is of the same degree as A051903(10) = 1, but then continuing, we have A003415(10) = 7, which is a prime, thus a(21) = 7.

Cf. A003415, A051674, A051903, A157037.

Cf. A003415, A051674, A051903, A157037, A328384 (gives the number of iterations needed to reach such a number).

Cf. A003415, A051674, A051903, A157037, A328384 (gives the number of iterations needed to reach such a number).

After the initial iteration of x -> A003415(x), starting from x=n, a(n) is the first number reached which is either a fixed point of arithmetic derivative (A003415), or a prime, or whose degree (A051903) differs from the degree of n. If n is of the form p^p, a(n) = n, otherwise a(n) is the first number found by iterating the map x -> A003415(x) that is different from n and either a prime, or whose degree (A051903) differs from the degree of n.

After the initial iteration of x -> A003415(x), starting from x=n, a(n) is the first number reached which is either a fixed point of arithmetic derivative (A003415), or a prime, or whose degree (A051903) differs from the degree of n. If n is of the form p^p, a(n) = n, otherwise a(n) is the first number found by iterating the map x -> A003415(x) that is different from n and either a prime, or whose degree (A051903) differs from the degree of n.

a(1) = 0, [as here the degrees of 0 and a(p) = 1 for all primesare considered different].

a(p) = 1 for all primes.

a(n) = A003415^(k)(n), when k = abs(A328384(n)). [Taking the abs(A328384(n))-th arithmetic derivative of n gives a(n)]

Cf. also A256750, A327968, A327975, A327977, A328248, A328252, A328305, A328310, A328311, A328320, A328321.

a(n) = A003415^(k)(n), when k = A328384(n). [Taking the A328384(n)-th arithmetic derivative of n gives a(n)]

Cf. A003415, A051674, A051903, A157037, A328384.

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