A260621 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.

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

Offset: 1

Author

Matthew Campbell, Sep 25 2015

Keywords

• nonn

Comments

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

Examples

			When n = 3, writing Composite(x) for A002808(x):
1. Composite(3) = 8. 8 + 1 = 9 = 3^2. 9 is not prime.
2. Composite(8) = 15. 15 + 1 = 16 = 2^4. 16 is not prime.
3. Composite(15) = 25. 25 + 1 = 26 = 2*13. 26 is not prime.
4. Composite(25) = 38. 38 + 1 = 39 = 3*13. 39 is not prime.
5. Composite(38) = 55. 55 + 1 = 56 = 2^3*7. 56 is not prime.
6. Composite(55) = 77. 77 + 1 = 78 = 2*3*13. 78 is not prime.
7. Composite(77) = 105. 105 + 1 = 106 = 2*53. 106 is not prime.
8. Composite(105) = 140. 140 + 1 = 141 = 3*47. 141 is not prime.
9. Composite(140) = 183. 183 + 1 = 184 = 2^3*23. 184 is not prime.
10. Composite(183) = 235. 235 + 1 = 236 = 2^2*59. 236 is not prime.
11. Composite(235) = 298. 298 + 1 = 299 = 13*23. 299 is not prime.
12. Composite(298) = 372. 372 + 1 = 373. 373 is prime.
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Since the composite function was applied 12 times, a(3)=12.
		

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Crossrefs

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

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

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

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

Composites with prime subscripts: A065858.

Composites without prime subscripts: A175251.

Order of compositeness: A059981, A236536.

Prime(n)-1: A006093.

Programs

Mathematica

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 *)

Extensions

Terms from a(12) onward from Jon E. Schoenfield, Sep 27 2015