%I #6 Oct 07 2013 03:28:02

%S 3,5,7,29,41,79,89,107,109,127,149,157,179,191,199,211

%N Primes that are conjectured to lead to a one-cycle for every natural number x in the following (nontrivial) generalization of the (3x+1) problem.

%C Start with a number x and construct a successor by the following iterative procedure: first remove all factors 2, 3, 5, ..., p(k) from x, where p(k) is the k-th prime number. When no further such factors remain then take the number ([p(k+1)*x]+1)/2 as the successor. The (3x+1) problem is the special case k=1 in the sequence that lists the p(k+1) leading to a one-cycle.

%C For other primes there is at least 1 supplementary cycle: e.g. when p(k+1)=11 there is also a cycle starting with 17; when p(k+1)=19 there is also a cycle starting with 46063; when p(k+1)=61 there are 3 supplementary cycles starting resp. with 97, 199, 26833; etc.

%t v[n_, k_]:=Block[{m=n}, Do[While[Mod[m, Prime[i]]==0, m=m/Prime[i]], {i, k}]; If[m!=1, Prepend[v[m*Prime[k+1]+1, k], m], v[m, k]={1}]] b[r_, s_, t_]:=Table[v[n, r], {n, s, t}]

%K nonn,more

%O 1,1

%A _Herman Roelants_, Oct 11 2004