This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A120686 #14 Aug 01 2024 10:37:15 %S A120686 5,10,15,19,20,25,30,31,38,40,45,47,50,53,57,60,61,62,75,76,80,90,93, %T A120686 94,95,97,100,103,106,109,114,120,122,124,125,133,135,141,149,150,152, %U A120686 155,159,160,163,171,173,180,183,186,188,190,191,194,199,200 %N A120686 Integers m such that the sequence defined by f(0)=m and f(n+1)=2+gpf(f(n)), with gpf(n) being the greatest prime factor of n (A006530), ends up in the period 3 cycle 5 -> 7 -> 9 -> 5 -> ... %C A120686 Let f(0)=m; f(n+1)= c + d gpf(f(n)), where gpf(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. %C A120686 Note that if f(n) is not prime then f(n+1)= 2 + gpf(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+gpf(2^k)=2+2=4. %e A120686 Oscillation between 5,7,9: 2+gpf(5)=2+5=7; 2+gpf(7)=2+7=9; 2+gpf(9)=2+3=5. %e A120686 Fixed point is 4: 2+gpf(4)=2+2=4. %t A120686 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 %Y A120686 Cf. A120685, A120684, A072268, A006530. %K A120686 nonn %O A120686 0,1 %A A120686 _Carlos Alves_, Jun 25 2006