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A374066 a(n) is the number of terms in the trajectory when the map x -> A067240(x) is iterated, starting from x = n until x = 0.

%I A374066 #23 Jun 30 2024 22:50:07
%S A374066 2,3,4,4,5,5,6,5,6,6,7,5,6,7,6,6,7,7,8,6,6,8,9,6,7,7,8,6,7,7,8,7,6,8,
%T A374066 7,6,7,9,8,6,7,7,8,6,7,10,11,7,8,7,8,8,9,9,8,7,7,8,9,6,7,9,6,8,7,7,8,
%U A374066 8,7,8,9,7,8,8,9,7,7,7,8,6,10,8,9,7,7,9,8,8,9
%N A374066 a(n) is the number of terms in the trajectory when the map x -> A067240(x) is iterated, starting from x = n until x = 0.
%e A374066 For n=11, the trajectory from n down to 0 comprises a(11) = 7 terms: 11 -> 10 -> 5 -> 4 -> 2 -> 1 -> 0.
%p A374066 f := proc(n)
%p A374066     local e, j:
%p A374066     e := ifactors(n)[2]:
%p A374066     add((e[j][1] - 1) * e[j][1]^(e[j][2] - 1), j = 1 .. nops(e))
%p A374066 end proc:
%p A374066  A374066:= proc(n)
%p A374066     local count, current:
%p A374066     count := 1:
%p A374066     current := n:
%p A374066     while current <> 0 do
%p A374066         current := f(current):
%p A374066         count := count + 1
%p A374066     end do:
%p A374066     return count
%p A374066 end proc:
%p A374066 map(A374066, [$1..200]);
%t A374066 f[p_, e_] := (p - 1)*p^(e - 1); s[n_] := s[n] = Plus @@ f @@@ FactorInteger[n]; a[n_] := Length[NestWhileList[s, n, # > 0 &]]; Array[a, 100] (* _Amiram Eldar_, Jun 27 2024 *)
%Y A374066 Cf. A000010, A067240.
%K A374066 nonn
%O A374066 1,1
%A A374066 _Rafik Khalfi_, Jun 27 2024