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

Author

Carlos Alves, Jun 25 2006

Keywords

• nonn

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.

Examples

			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).
		

Crossrefs

Cf. A072268, A006530.

Programs

Mathematica

f = Function[n, FactorInteger[n][[ -1, 1]] + 1]; mn = Map[(NestList[f, #, 8][[ -1]]) &, Range[2, 500]]; out = Flatten[Position[mn, 3]] + 1

Extensions

Edited by Michel Marcus, Feb 23 2013