A234694 - OEIS

%I #15 Apr 05 2014 22:31:43

%S 0,1,0,2,1,2,1,0,0,2,2,4,1,1,2,4,2,1,1,2,3,3,2,3,1,1,1,3,5,4,3,4,3,3,

%T 3,2,4,3,2,5,4,4,4,1,1,5,4,2,1,2,5,5,2,3,4,2,3,5,7,7,6,2,5,6,2,5,4,4,

%U 7,6,6,5,4,8,7,4,5,3,5,7,3,5,4,7,6,7,2

%N a(n) = |{0 < k < n: p = k + prime(n-k) and prime(p) - p + 1 are both prime}|.

%C Conjecture: (i) a(n) > 0 for all n > 9. Also, for any integer n > 51 there is a positive integer k < n such that p = k + prime(n-k) and prime(p) + p + 1 are both prime.

%C (ii) If n > 9 (or n > 21), then there is a positive integer k < n such that m - 1 and prime(m) + m (or prime(m) - m, resp.) are both prime, where m = k + prime(n-k).

%C (iii) If n > 483, then for some 0 < k < n both prime(m) + m and prime(m) - m are prime, where m = k + prime(n-k).

%C (iv) If n > 3, then there is a positive integer k < n such that prime(k + prime(n-k)) + 2 is prime.

%C Clearly, part (i) of the conjecture implies that there are infinitely many primes p with prime(p) - p + 1 (or prime(p) + p + 1) also prime.

%C See A234695 for primes p with prime(p) - p + 1 also prime.

%H Zhi-Wei Sun, <a href="/A234694/b234694.txt">Table of n, a(n) for n = 1..10000</a>

%H Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014

%e a(5) = 1 since 2 + prime(3) = 7 and prime(7) - 6 = 11 are both prime.

%e a(25) = 1 since 20 + prime(5) = 31 and prime(31) - 30 = 97 are both prime.

%e a(27) = 1 since 18 + prime(9) = 41 and prime(41) - 40 = 139 are both prime.

%e a(45) = 1 since 6 + prime(39) = 173 and prime(173) - 172 = 859 are both prime.

%e a(49) = 1 since 26 + prime(23) = 109 and prime(109) - 108 = 491 are both prime.

%t f[n_,k_]:=k+Prime[n-k]

%t q[n_,k_]:=PrimeQ[f[n,k]]&&PrimeQ[Prime[f[n,k]]-f[n,k]+1]

%t a[n_]:=Sum[If[q[n,k],1,0],{k,1,n-1}]

%t Table[a[n],{n,1,100}]

%Y Cf. A000040, A014688, A014689, A014692, A064269, A064270, A232861, A233150, A233183, A233206, A233296, A234695.

%K nonn

%O 1,4

%A _Zhi-Wei Sun_, Dec 29 2013