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A347388 The first 3-smooth number eventually reached when iterating Dedekind psi function from n, with a(n) = n if n is already a 3-smooth number.

%I A347388 #10 Aug 31 2021 19:10:39
%S A347388 1,2,3,4,6,6,8,8,9,18,12,12,24,24,24,16,18,18,36,36,32,36,24,24,72,96,
%T A347388 27,48,72,72,32,32,48,54,48,36,144,144,96,72,96,96,72,72,72,72,48,48,
%U A347388 96,216,72,192,54,54,72,96,144,216,144,144,96,96,96,64,192,144,108,108,96,144,72,72,576,576,288,288,96,384
%N A347388 The first 3-smooth number eventually reached when iterating Dedekind psi function from n, with a(n) = n if n is already a 3-smooth number.
%C A347388 See comments and references in A019269, which gives the number of iterations needed to reach a(n).
%F A347388 If A006530(n) <= 3, then a(n) = n, otherwise a(n) = a(A001615(n)).
%t A347388 psi[1] = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); a[n_] := NestWhile[psi, n, FactorInteger[#][[-1, 1]] > 3 &]; Array[a, 100] (* _Amiram Eldar_, Aug 31 2021 *)
%o A347388 (PARI)
%o A347388 A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
%o A347388 A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
%o A347388 A347388(n) = if(A006530(n)<=3,n,A347388(A001615(n)));
%Y A347388 Cf. A001615, A003586, A006530, A019269, A065333.
%Y A347388 Cf. also A347387.
%K A347388 nonn
%O A347388 1,2
%A A347388 _Antti Karttunen_, Aug 31 2021