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

%I A383619 #21 May 14 2025 17:49:32
%S A383619 1,8,20,36,40,54,64,84,104,116,135,144,196,224,228,232,243,264,270,280
%N A383619 Conjectured list of least elements of nontrivial arithmetic derivative orbits.
%C A383619 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.
%C A383619 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.
%C A383619 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.
%F A383619 Let {C_n} be the family of nontrivial equivalence classes under iteration of the arithmetic derivative operator. Then, a(n) := min(C_n).
%e A383619 a(2) := min(C_2) = 8.
%o A383619 (SageMath)
%o A383619 D = lambda n: 0 if n<2 else sum(e*(n//p) for p,e in Integer(n).factor())
%o A383619 def A(N, k=None):
%o A383619     c, o = {1:1}, [1]
%o A383619     for i in range(2, N+1):
%o A383619         if i in c: continue
%o A383619         P, m = [], i
%o A383619         while 1 <= m <= N and m not in c and m not in P:
%o A383619             P.append(m)
%o A383619             m = D(m)
%o A383619         if   m in c:      v = c[m]
%o A383619         elif m in P:      v = min(P[P.index(m):])
%o A383619         else:             v = min(P)
%o A383619         for x in P: c[x] = v
%o A383619         if c[i] == i and len(P) > 1:
%o A383619             o.append(i)
%o A383619             if k and len(o) >= k: break
%o A383619     return o
%o A383619 A(10**18, k=20)
%Y A383619 Cf. A003415, A068346, A099306, A258644, A258645, A258646, A258647, A258648, A258649, A258650.
%K A383619 nonn,more
%O A383619 1,2
%A A383619 _Dimitris Cardaris_, May 02 2025