A260621 - OEIS

%I #52 Jul 17 2016 09:25:44

%S 1,1,12,2,1,1,3,11,1,1,7,9,1,2,10,4,2,1,1,6,8,3,3,1,9,3,1,1,18,3,1,5,

%T 7,2,2,1,4,8,2,14,1,1,6,17,2,6,1,4,6,1,1,2,2,3,7,1,13,6,1,4,16,5,16,1,

%U 5,31,35,3,5,2,1,2,3,1,1,2,6,1,1,12,5,1,2

%N Let b(k, n) = number obtained when the map x->A002808(x) is applied k times to n; a(n) is the smallest k such that b(k, n) + 1 is prime.

%C a(n) is also the smallest value of k at which b(k, n+1) - b(k, n) > 1.

%H Jon E. Schoenfield, <a href="/A260621/b260621.txt">Table of n, a(n) for n = 1..10000</a>

%e When n = 3, writing Composite(x) for A002808(x):

%e 1. Composite(3) = 8. 8 + 1 = 9 = 3^2. 9 is not prime.

%e 2. Composite(8) = 15. 15 + 1 = 16 = 2^4. 16 is not prime.

%e 3. Composite(15) = 25. 25 + 1 = 26 = 2*13. 26 is not prime.

%e 4. Composite(25) = 38. 38 + 1 = 39 = 3*13. 39 is not prime.

%e 5. Composite(38) = 55. 55 + 1 = 56 = 2^3*7. 56 is not prime.

%e 6. Composite(55) = 77. 77 + 1 = 78 = 2*3*13. 78 is not prime.

%e 7. Composite(77) = 105. 105 + 1 = 106 = 2*53. 106 is not prime.

%e 8. Composite(105) = 140. 140 + 1 = 141 = 3*47. 141 is not prime.

%e 9. Composite(140) = 183. 183 + 1 = 184 = 2^3*23. 184 is not prime.

%e 10. Composite(183) = 235. 235 + 1 = 236 = 2^2*59. 236 is not prime.

%e 11. Composite(235) = 298. 298 + 1 = 299 = 13*23. 299 is not prime.

%e 12. Composite(298) = 372. 372 + 1 = 373. 373 is prime.

%e --------------------------------------------------------------

%e Since the composite function was applied 12 times, a(3)=12.

%t c = Select[Range[10^5], CompositeQ]; Table[k = 1; While[! PrimeQ[Nest[c[[#]] &, n, k] + 1], k++]; k, {n, 120}] (* _Michael De Vlieger_, Jul 15 2016 *)

%Y Primes and nonprimes: A000040, A002808, A008578, A018252.

%Y a(1) = p, a(n+1) = a(n)-th composite number: A006508, A022450, A022451, A025010, A025011, A059407, A059408.

%Y Composites with order n > 1: A050435, A050436, A050438, A050439, A050440.

%Y Composites with order n = b, n >= 1: A022449.

%Y Composites with prime subscripts: A065858.

%Y Composites without prime subscripts: A175251.

%Y Order of compositeness: A059981, A236536.

%Y Prime(n)-1: A006093.

%K nonn

%O 1,3

%A _Matthew Campbell_, Sep 25 2015

%E Terms from a(12) onward from _Jon E. Schoenfield_, Sep 27 2015