A120685 Integers m such that the sequence defined by f(0)=m and f(n+1)=1+gpf(f(n)), with gpf(n) being the greatest prime factor of n (A006530), ends up in the repetitive cycle 4 -> 3 -> 4 -> ...

2, 4, 5, 8, 10, 11, 13, 15, 16, 17, 20, 22, 23, 25, 26, 30, 32, 33, 34, 37, 39, 40, 41, 44, 45, 46, 47, 50, 51, 52, 53, 55, 60, 61, 64, 65, 66, 68, 69, 71, 74, 75, 77, 78, 80, 82, 83, 85, 88, 90, 91, 92, 94, 97, 99, 100, 102, 104, 106, 107, 110, 111, 113, 115, 117, 119, 120

Offset

0,1

Comments

Let f(0)=m; f(n+1)=1+gpf(f(n)), where gpf(n) is the greatest prime factor of n (A006530). For any m, for sufficiently large n the sequence f(n) oscillates between 3 and 4. Given a sufficiently large n, this allows us to divide integers in two classes: C3 (m such that the sequence f(n) enters the cycle 3, 4, 3, ...) and C4 (m such that the sequence f(n) enters the cycle 4, 3, 4, ...). We present here C4 as the one that begin with 4. In A120684 we present C3 as the one that begin with 3.

Links

Table of n, a(n) for n=0..66.

Example

Oscillation between 3 and 4: 1+gpf(3)=1+3=4; 1+gpf(4)=1+2=3.
Other value, e.g. 7: 1+gpf(7)=1+7=8; 1+gpf(8)=1+2=3 (7 belongs to C3).
Other value, e.g. 20: 1+gpf(20)=1+5=6; 1+gpf(6)=1+3=4 (20 belongs to C4).

Mathematica

f = Function[n, FactorInteger[n][[ -1, 1]] + 1]; mn = Map[(NestList[f, #, 8][[ -1]]) &, Range[2, 500]]; out = Flatten[Position[mn, 4]] + 1

Crossrefs

Cf. A120684, A072268, A006530.

Sequence in context: A026138 A026170 A026174 * A284472 A160545 A161790

Adjacent sequences: A120682 A120683 A120684 * A120686 A120687 A120688

Keyword

nonn

Author

Carlos Alves, Jun 25 2006

Extensions

Edited by Michel Marcus, Feb 25 2013

Status

approved