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.

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, 27, 48, 72, 72, 32, 32, 48, 54, 48, 36, 144, 144, 96, 72, 96, 96, 72, 72, 72, 72, 48, 48, 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

Offset: 1

Author

Antti Karttunen, Aug 31 2021

Keywords

• nonn

Comments

See comments and references in A019269, which gives the number of iterations needed to reach a(n).

Crossrefs

Cf. A001615, A003586, A006530, A019269, A065333.

Cf. also A347387.

Programs

Mathematica

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 *)

PARI

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
A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
A347388(n) = if(A006530(n)<=3,n,A347388(A001615(n)));

Formula

If A006530(n) <= 3, then a(n) = n, otherwise a(n) = a(A001615(n)).