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A323135 a(n) is the least number of iterations that n requires to reach a power of a prime under the map x -> A070229(x), or -1 if we never reach a power of a prime.

%I A323135 #16 Apr 06 2020 08:05:40
%S A323135 0,0,0,0,0,1,0,0,0,3,0,3,0,5,2,0,0,5,0,1,4,9,0,1,0,11,0,3,0,3,0,0,8,
%T A323135 15,2,11,0,17,10,9,0,1,0,7,8,21,0,15,0,7,14,9,0,17,6,7,16,27,0,9,0,29,
%U A323135 6,0,8,5,0,13,20,5,0,15,0,35,14,15,4,7,0,13,0
%N A323135 a(n) is the least number of iterations that n requires to reach a power of a prime under the map x -> A070229(x), or -1 if we never reach a power of a prime.
%C A323135 The powers of primes correspond to A000961.
%C A323135 Apparently, a(n) >= 0 for any n > 0.
%C A323135 For a given number n, while iterating A070229, we may encounter several prime increments (=several greatest prime factors). It is likely that the number of distinct increments before reaching a power of a prime is not bounded.
%C A323135 For k = 0..9, the least numbers with k distinct increments are:
%C A323135   k  n       Increments
%C A323135   -  ------  ------------------------------------------
%C A323135   0       1  {}
%C A323135   1       6  {3}
%C A323135   2      12  {3, 5}
%C A323135   3      72  {3, 5, 17}
%C A323135   4     135  {5, 7, 11, 17}
%C A323135   5     686  {7, 11, 13, 19, 41}
%C A323135   6   12408  {47, 53, 59, 71, 89, 149}
%C A323135   7   35378  {19, 23, 67, 89, 101, 179, 211}
%C A323135   8  127581  {43, 53, 73, 103, 113, 227, 283, 457}
%C A323135   9  222111  {37, 79, 97, 191, 233, 239, 311, 359, 631}
%H A323135 Rémy Sigrist, <a href="/A323135/b323135.txt">Table of n, a(n) for n = 1..10000</a>
%F A323135 a(n) = 0 iff n belongs to A000961.
%e A323135 For n = 12:
%e A323135 - 12 = 2^2 * 3,
%e A323135 - A070229(12) = 12 + 3 = 15 = 3 * 5,
%e A323135 - A070229(15) = 15 + 5 = 20 = 2^2 * 5,
%e A323135 - A070229(20) = 20 + 5 = 25 = 5^5,
%e A323135 - hence we need at least 3 iterations of A070229 to reach a power of a prime,
%e A323135 - and a(12) = 3.
%o A323135 (PARI) a(n) = for (k=0, oo, if (omega(n) <= 1, return (k), my (f=factor(n)); n += f[#f~, 1]))
%Y A323135 See A323136 for the corresponding powers of primes.
%Y A323135 Cf. A000961, A070229.
%K A323135 nonn,look
%O A323135 1,10
%A A323135 _Rémy Sigrist_, Jan 05 2019