A340324 Numbers k such that starting with k and repeatedly applying the map x -> A340323(x) reaches the loop {3, 4}.

2, 3, 4, 7, 8, 9, 16, 21, 27, 31, 32, 63, 64, 81, 93, 127, 128, 189, 217, 243, 256, 279, 381, 512, 567, 651, 729, 837, 889, 1024, 1143, 1701, 1953, 2048, 2187, 2511, 2667, 3429, 3937, 4096, 5103, 5859, 6561, 7533, 8001, 8191, 8192, 10287, 11811, 15309, 16384

Offset: 1

Author

José María Grau Ribas, Jan 05 2021

Keywords

• nonn

Comments

From Sebastian Karlsson, Jan 15 2021: (Start)

The sequence can be defined exclusively as:

- Powers of two greater than one.

- Powers of three greater than one.

- Products of distinct Mersenne primes (A046528, except initial 1) or powers of three multiplied with products of distinct Mersenne primes. (End)

Crossrefs

Cf. A340323, A340325.

Cf. A000079, A000244, A000668, A046528.

Programs

Mathematica

fa[n_]:=fa[n]=FactorInteger[n]; phi[1]=1; phi[p_, s_]:= (p + 1)*( p - 1)^(s - 1)
phi[n_]:=Product[phi[fa[n][[i, 1]], fa[n][[i, 2]]], {i, Length[fa[n]]}];
S[n_] := NestWhile [phi, n, ! ( # == 12 || # == 3 || # == 4) &];
Select[1 + Range[100], 2
				

PARI

f(n) = if(1==n, n, my(f=factor(n)); prod(i=1, #f~, (f[i, 1]+1)*((f[i, 1]-1)^(f[i, 2]-1)))); \\ A340323
isok(m) = if (m==1, return(0)); while(! ((m==3) || (m==4) || (m==12)), m = f(m)); ((m==3) || (m==4)); \\ Michel Marcus, Jan 21 2021