A120684 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 3 -> 4 -> 3 -> ...

3, 6, 7, 9, 12, 14, 18, 19, 21, 24, 27, 28, 29, 31, 35, 36, 38, 42, 43, 48, 49, 54, 56, 57, 58, 59, 62, 63, 67, 70, 72, 73, 76, 79, 81, 84, 86, 87, 89, 93, 95, 96, 98, 101, 103, 105, 108, 109, 112, 114, 116, 118, 124, 126, 127, 129, 131, 133, 134, 137, 140, 144, 145, 146

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 C3 as the one that begin with 3. In A120685 we present C4 as the one that begin with 4.

Links

Table of n, a(n) for n=0..63.

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, 3]] + 1

Crossrefs

Cf. A072268, A006530.

Sequence in context: A176409 A376088 A284473 * A324927 A026227 A026232

Adjacent sequences: A120681 A120682 A120683 * A120685 A120686 A120687

Keyword

nonn

Author

Carlos Alves, Jun 25 2006

Extensions

Edited by Michel Marcus, Feb 23 2013

Status

approved