|
|
A159836
|
|
Integers n such that the orbit n, f(n), f(f(n)), ... is eventually periodic with period 2, where f(n) = product(a(k)^p(k)) when n has the prime factorization n = product(p(k)^a(k)).
|
|
2
|
|
|
8, 9, 18, 24, 25, 32, 36, 40, 45, 49, 50, 56, 63, 64, 75, 81, 88, 90, 96, 98, 99, 100, 104, 117, 120, 121, 125, 126, 128, 136, 144, 147, 150, 152, 153, 160, 162, 168, 169, 171, 175, 180, 184, 192, 196, 198, 200, 207, 216, 224, 225, 232, 234, 242, 243, 245, 248
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
It is proved in the reference that for every positive integer n the orbit n, f(n), f(f(n)), ... is eventually periodic with period 1 or 2.
Includes all numbers whose prime exponents are distinct primes. If n is in this sequence and k is a squarefree number such that (k,n) = 1, then k*n is in this sequence. - Charlie Neder, May 16 2019
|
|
LINKS
|
|
|
MAPLE
|
with(numtheory); P:=proc(q) local a0f, a1, a1f, a2, a2f, a3, a3f, a4, a4f, k, n;
for n from 1 to q do a0:=1; a1:=1; a2:=2; a3:=3; a4:=n;
while not (a1=a3 and a2=a4) do a0f:=ifactors(a4)[2];
a1:=mul(a0f[k][2]^a0f[k][1], k=1..nops(a0f)); a1f:=ifactors(a1)[2];
a2:=mul(a1f[k][2]^a1f[k][1], k=1..nops(a1f)); a2f:=ifactors(a2)[2];
a3:=mul(a2f[k][2]^a2f[k][1], k=1..nops(a2f)); a3f:=ifactors(a3)[2];
a4:=mul(a3f[k][2]^a3f[k][1], k=1..nops(a3f)); od;
if a1<>a2 then print(n); fi; od; end: P(10^6); # Paolo P. Lava, Oct 24 2013
|
|
MATHEMATICA
|
f[n_] := Module[{f = Transpose[FactorInteger[n]]}, Times @@ (f[[2]]^f[[1]])]; Select[Range[300], (x = NestWhileList[f, #, UnsameQ, All]; x[[-2]] != x[[-1]]) &] (* T. D. Noe, Oct 24 2013 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|