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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. 2
 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, 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, 6, 0, 8, 5, 0, 13, 20, 5, 0, 15, 0, 35, 14, 15, 4, 7, 0, 13, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,10 COMMENTS The powers of primes correspond to A000961. Apparently, a(n) >= 0 for any n > 0. 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. For k = 0..9, the least numbers with k distinct increments are:   k  n       Increments   -  ------  ------------------------------------------   0       1  {}   1       6  {3}   2      12  {3, 5}   3      72  {3, 5, 17}   4     135  {5, 7, 11, 17}   5     686  {7, 11, 13, 19, 41}   6   12408  {47, 53, 59, 71, 89, 149}   7   35378  {19, 23, 67, 89, 101, 179, 211}   8  127581  {43, 53, 73, 103, 113, 227, 283, 457}   9  222111  {37, 79, 97, 191, 233, 239, 311, 359, 631} LINKS Rémy Sigrist, Table of n, a(n) for n = 1..10000 FORMULA a(n) = 0 iff n belongs to A000961. EXAMPLE For n = 12: - 12 = 2^2 * 3, - A070229(12) = 12 + 3 = 15 = 3 * 5, - A070229(15) = 15 + 5 = 20 = 2^2 * 5, - A070229(20) = 20 + 5 = 25 = 5^5, - hence we need at least 3 iterations of A070229 to reach a power of a prime, - and a(12) = 3. PROG (PARI) a(n) = for (k=0, oo, if (omega(n) <= 1, return (k), my (f=factor(n)); n += f[#f~, 1])) CROSSREFS See A323136 for the corresponding powers of primes. Cf. A000961, A070229. Sequence in context: A291971 A240923 A272727 * A100258 A045763 A132748 Adjacent sequences:  A323132 A323133 A323134 * A323136 A323137 A323138 KEYWORD nonn AUTHOR Rémy Sigrist, Jan 05 2019 STATUS approved

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Last modified June 16 06:46 EDT 2019. Contains 324145 sequences. (Running on oeis4.)