A236066 - OEIS

%I #6 Jan 18 2014 20:38:46

%S 5,98893,1110709,4231849,5319707,6763349,7904087,10823431,13893109,

%T 15323939,15544079,15716713,17642899,18978439,20126237

%N 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.

%C 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.

%C This is similar to the conjecture in A235925.

%H Zhi-Wei Sun, <a href="/A236066/b236066.txt">Table of n, a(n) for n = 1..15</a>

%e 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.

%e a(2) = 98893 with 98893, g(98893) = 1185113, g(1185113) = 17381209, g(17381209) = 304696943, g(304696943) = 6262760333, g(6262760333) = 148561011217 all prime.

%t g[n_]:=Prime[n]-n-1

%t 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]]]]]]]

%t n=0;Do[If[p[k],n=n+1;Print[n," ",Prime[k]]],{k,1,10^6}]

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

%K nonn

%O 1,1

%A _Zhi-Wei Sun_, Jan 18 2014