A323135 - OEIS

%I #16 Apr 06 2020 08:05:40

%S 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 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 6,0,8,5,0,13,20,5,0,15,0,35,14,15,4,7,0,13,0

%N 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 The powers of primes correspond to A000961.

%C Apparently, a(n) >= 0 for any n > 0.

%C 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 For k = 0..9, the least numbers with k distinct increments are:

%C   k  n       Increments

%C   -  ------  ------------------------------------------

%C   0       1  {}

%C   1       6  {3}

%C   2      12  {3, 5}

%C   3      72  {3, 5, 17}

%C   4     135  {5, 7, 11, 17}

%C   5     686  {7, 11, 13, 19, 41}

%C   6   12408  {47, 53, 59, 71, 89, 149}

%C   7   35378  {19, 23, 67, 89, 101, 179, 211}

%C   8  127581  {43, 53, 73, 103, 113, 227, 283, 457}

%C   9  222111  {37, 79, 97, 191, 233, 239, 311, 359, 631}

%H Rémy Sigrist, <a href="/A323135/b323135.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = 0 iff n belongs to A000961.

%e For n = 12:

%e - 12 = 2^2 * 3,

%e - A070229(12) = 12 + 3 = 15 = 3 * 5,

%e - A070229(15) = 15 + 5 = 20 = 2^2 * 5,

%e - A070229(20) = 20 + 5 = 25 = 5^5,

%e - hence we need at least 3 iterations of A070229 to reach a power of a prime,

%e - and a(12) = 3.

%o (PARI) a(n) = for (k=0, oo, if (omega(n) <= 1, return (k), my (f=factor(n)); n += f[#f~, 1]))

%Y See A323136 for the corresponding powers of primes.

%Y Cf. A000961, A070229.

%K nonn,look

%O 1,10

%A _Rémy Sigrist_, Jan 05 2019