

A120688


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 we present C7 as the one that includes 7. In A120688 (here) we present C9 as the one that includes 9.


3



3, 6, 9, 12, 13, 17, 18, 23, 24, 26, 27, 29, 34, 36, 39, 43, 46, 48, 51, 52, 54, 58, 59, 65, 68, 69, 72, 73, 78, 81, 85, 86, 87, 91, 92, 96, 101, 102, 104, 107, 108, 115, 116, 117, 118, 119, 129, 130, 131, 136, 138, 139, 143, 144, 145, 146, 153, 156, 157, 161, 162
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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..60.


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; Cc9


CROSSREFS

Cf. A120687, A120686, A120684, A072268, A006530.
Sequence in context: A257220 A092452 A230215 * A102014 A168045 A288522
Adjacent sequences: A120685 A120686 A120687 * A120689 A120690 A120691


KEYWORD

nonn


AUTHOR

Carlos Alves, Jun 25 2006


STATUS

approved



