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.

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, 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, 8, 7, 8, 9, 7, 8, 8, 9, 7, 7, 7, 8, 6, 10, 8, 9, 7, 7, 9, 8, 8, 9

Offset: 1

Author

Rafik Khalfi, Jun 27 2024

Keywords

• nonn

Examples

			For n=11, the trajectory from n down to 0 comprises a(11) = 7 terms: 11 -> 10 -> 5 -> 4 -> 2 -> 1 -> 0.
		

Crossrefs

Cf. A000010, A067240.

Programs

Maple

f := proc(n)
    local e, j:
    e := ifactors(n)[2]:
    add((e[j][1] - 1) * e[j][1]^(e[j][2] - 1), j = 1 .. nops(e))
end proc:
 A374066:= proc(n)
    local count, current:
    count := 1:
    current := n:
    while current <> 0 do
        current := f(current):
        count := count + 1
    end do:
    return count
end proc:
map(A374066, [$1..200]);

Mathematica

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 *)