A236066 Primes p with g(p), g(g(p)), g(g(g(p))), g(g(g(g(p)))), g(g(g(g(g(p))))) all prime, where g(n) = prime(n) - n - 1.

5, 98893, 1110709, 4231849, 5319707, 6763349, 7904087, 10823431, 13893109, 15323939, 15544079, 15716713, 17642899, 18978439, 20126237

Offset: 1

Author

Zhi-Wei Sun, Jan 18 2014

Keywords

• nonn

Comments

Conjecture: For any integer m > 1, there are infinitely many chains p(1) < ... < p(m) of m primes with p(k+1) = prime(p(k)) - p(k) - 1 for all 0 < k < m.

This is similar to the conjecture in A235925.

Examples

			a(1) = 5 since neither g(2) = prime(2) - 2 - 1 = 0 nor g(3) = prime(3) - 3 - 1 = 1 is prime, but 5 = g(5) = g(g(5)) =  g(g(g(5))) = g(g(g(g(5)))) = g(g(g(g(g(5))))) is prime.
a(2) = 98893 with 98893, g(98893) = 1185113, g(1185113) = 17381209, g(17381209) = 304696943, g(304696943) = 6262760333, g(6262760333) = 148561011217 all prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..15

Crossrefs

Cf. A000040, A234695, A235925, A235934, A235935, A235984.

Programs

Mathematica

g[n_]:=Prime[n]-n-1
p[k_]:=PrimeQ[g[Prime[k]]]&&PrimeQ[g[g[Prime[k]]]]&&PrimeQ[g[g[g[Prime[k]]]]]&&PrimeQ[g[g[g[g[Prime[k]]]]]]&&PrimeQ[g[g[g[g[g[Prime[k]]]]]]]
n=0;Do[If[p[k],n=n+1;Print[n," ",Prime[k]]],{k,1,10^6}]