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A120686 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 (here) we present C5 as the one that includes 5. In A120687 we present C7 as the one that includes 7. In A120688 we present C9 as the one that includes 9. 3
5, 10, 15, 19, 20, 25, 30, 31, 38, 40, 45, 47, 50, 53, 57, 60, 61, 62, 75, 76, 80, 90, 93, 94, 95, 97, 100, 103, 106, 109, 114, 120, 122, 124, 125, 133, 135, 141, 149, 150, 152, 155, 159, 160, 163, 171, 173, 180, 183, 186, 188, 190, 191, 194, 199, 200 (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..55.

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; Cc5

CROSSREFS

Cf. A120685, A120684, A072268, A006530.

Sequence in context: A344958 A313677 A241141 * A313678 A313679 A158635

Adjacent sequences:  A120683 A120684 A120685 * A120687 A120688 A120689

KEYWORD

nonn

AUTHOR

Carlos Alves, Jun 25 2006

STATUS

approved

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Last modified October 23 10:54 EDT 2021. Contains 348211 sequences. (Running on oeis4.)