A099306 -id:A099306 - cp's OEIS frontend

A383619 Conjectured list of least elements of nontrivial arithmetic derivative orbits.

1, 8, 20, 36, 40, 54, 64, 84, 104, 116, 135, 144, 196, 224, 228, 232, 243, 264, 270, 280

Offset: 1

Dimitris Cardaris, May 02 2025

a(n) is the least integer in the n-th nontrivial equivalence class under iteration of the Lagarias arithmetic derivative D. Two numbers m and n lie in the same class if repeated application of D to each, eventually produces the same value; every such class has a unique smallest member, and this sequence lists those members in ascending order.
Note that k is a fixed point of the arithmetic derivative D (i.e., D(k)=k) if and only if k=p^p for some prime p. Such one-element classes {p^p} are considered trivial and are excluded from the list of nontrivial attractors.
The values of a(n) are conjectural, contingent on the absence of further merges or the existence of nontrivial cycles beyond the computational horizon; this is analogous to Collatz dynamics.

			a(2) := min(C_2) = 8.

Cf. A003415, A068346, A099306, A258644, A258645, A258646, A258647, A258648, A258649, A258650.

D = lambda n: 0 if n<2 else sum(e*(n//p) for p,e in Integer(n).factor())
def A(N, k=None):
    c, o = {1:1}, [1]
    for i in range(2, N+1):
        if i in c: continue
        P, m = [], i
        while 1 <= m <= N and m not in c and m not in P:
            P.append(m)
            m = D(m)
        if   m in c:      v = c[m]
        elif m in P:      v = min(P[P.index(m):])
        else:             v = min(P)
        for x in P: c[x] = v
        if c[i] == i and len(P) > 1:
            o.append(i)
            if k and len(o) >= k: break
    return o
A(10**18, k=20)

Let {C_n} be the family of nontrivial equivalence classes under iteration of the arithmetic derivative operator. Then, a(n) := min(C_n).

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A383619 Conjectured list of least elements of nontrivial arithmetic derivative orbits.

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