

A120687


Let f(0)=m; f(n+1)= c + d lpf(f(n)), where lpf(n) is the largest prime factor of n (A006530). For any m, for sufficiently large n the sequence f(n) oscillates. In A120684,A120685 the values d=c=1 were considered. Here we consider d=1, c=2 and this allows us to divide integers in 4 classes: C4 (m such that f(n)=4, which is a fixed point); C5 (m such that f(n)=5, then oscillates between 5,7,9); C7 (m such that f(n)=7, then oscillates between 7,9,5); C9 (m such that f(n)=9, then oscillates between 9,5,7); In A120686 we present C5 as the one that includes 5. In A120687 (here) we present C7 as the one that includes 7. In A120688 we present C9 as the one that includes 9.


3



7, 11, 14, 21, 22, 28, 33, 35, 37, 41, 42, 44, 49, 55, 56, 63, 66, 67, 70, 71, 74, 77, 79, 82, 83, 84, 88, 89, 98, 99, 105, 110, 111, 112, 113, 121, 123, 126, 127, 132, 134, 137, 140, 142, 147, 148, 151, 154, 158, 164, 165, 166, 167, 168, 175, 176, 178, 179, 185, 189
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,1


COMMENTS

Note that if f(n) is not prime then f(n+1)= 2 + lpf(f(n)) <= 2 + f(n)/2 and the sequence decreases. If f(n) is prime and 2+f(n) is prime, the sequence will decrease when 2k+f(n) is not prime, which must occur for k>2. The bottom limit case is the cycle (5 7 9). The only other possibility occurs for 2^k numbers that go to the fixed point 4 because 2+lpf(2^k)=2+2=4.


LINKS

Table of n, a(n) for n=0..59.


EXAMPLE

Oscillation between 5,7,9:
2+lpf(5)=2+5=7; 2+lpf(7)=2+7=9; 2+lpf(9)=2+3=5.
Fixed point is 4: 2+lpf(4)=2+2=4.


MATHEMATICA

fi = Function[n, FactorInteger[n][[ 1, 1]] + 2]; mn = Map[(NestList[fi, #, 6][[ 1]]) &, Range[2, 200]]; Cc4 = Flatten[Position[mn, 4]] + 1; Cc5 = Flatten[Position[mn, 5]] + 1; Cc7 = Flatten[Position[mn, 7]] + 1; Cc9 = Flatten[Position[mn, 9]] + 1; Cc7


CROSSREFS

Cf. A120686, A120684, A072268, A006530.
Sequence in context: A080837 A168135 A225858 * A194468 A053217 A144931
Adjacent sequences: A120684 A120685 A120686 * A120688 A120689 A120690


KEYWORD

nonn


AUTHOR

Carlos Alves, Jun 25 2006


STATUS

approved



