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A087717 Start with x=n, then iterate the map x -> A322982(x) with A322982(x)=2*x-1 if x is noncomposite, otherwise A322982(x) = A032742(x), the largest proper divisor of x. If this iteration leads to a fixed point then a(n) is the value of that fixed point. If the iteration leads to a cycle, a(n) is the smallest value in the cycle. If the iteration never becomes periodic then a(n)=0.

%I A087717 #12 Jan 04 2019 09:03:10
%S A087717 1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,19,3,3,3,3,3,3,3,3,3,19,3,3,3,3,
%T A087717 3,3,3,19,19,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,19,19,3,3,3,3,3,3,3,
%U A087717 3,19,3,3,3,3,3,19,19,3,19,3,3,3,3,3,3,3,3,3,3,19,3,3,3,3,3,3,3,19,3,3,3,3,3
%N A087717 Start with x=n, then iterate the map x -> A322982(x) with A322982(x)=2*x-1 if x is noncomposite, otherwise A322982(x) = A032742(x), the largest proper divisor of x. If this iteration leads to a fixed point then a(n) is the value of that fixed point. If the iteration leads to a cycle, a(n) is the smallest value in the cycle. If the iteration never becomes periodic then a(n)=0.
%C A087717 Conjecture. For n > 1, the iteration given in the definition above always leads to the 3-cycle {3,5,9,3} or the 6-cycle {19,37,73,145,29,57,19}, thus a(n) takes on only the values 3 or 19 for n=2,3,4,.... This has been verified to n=1000000.
%C A087717 In range 2..100000 term 3 occurs 77630 times, while 19 occurs 22369 times. - _Antti Karttunen_, Jan 03 2019
%H A087717 Antti Karttunen, <a href="/A087717/b087717.txt">Table of n, a(n) for n = 1..16384</a>
%H A087717 Antti Karttunen, <a href="/A087717/a087717.txt">Data supplement: n, a(n) computed for n = 1..100000</a>
%t A087717 Which[Length@ Union@ #[[-2 ;; -1]] == 1, Last@ #, MemberQ[{3, 5, 9}, Last@ #], 3, MemberQ[{19, 37, 73, 145, 29, 57}, Last@ #], 19, True, 0] & /@ Array[NestWhileList[If[CompositeQ@ #, Divisors[#][[-2]], 2 # - 1] &, #, UnsameQ[##] &, All] &, 106] (* _Michael De Vlieger_, Jan 03 2019 *)
%o A087717 (PARI)
%o A087717 A322982(n) = if((1==n)||isprime(n),n+n-1,n/vecmin(factor(n)[,1]));
%o A087717 A087717(n) = { my(visited = Map(), visited_at_step = Map(), j=0, m=0, t); while(!mapisdefined(visited, n), mapput(visited, n, j); mapput(visited_at_step, j, n); j++; n = A322982(n)); for(k=mapget(visited,n), j-1, t = mapget(visited_at_step,k); if(!m || (t<m), m=t)); (m); }; \\ _Antti Karttunen_, Jan 03 2019
%Y A087717 Cf. A322982.
%K A087717 nonn
%O A087717 1,2
%A A087717 _John W. Layman_, Sep 29 2003
%E A087717 Name edited and the term a(1) = 1 prepended by _Antti Karttunen_, Jan 03 2019