A237712 - OEIS

%I #17 Apr 06 2014 22:20:57

%S 0,1,1,1,0,1,3,1,3,1,2,3,4,3,3,2,2,4,4,1,5,2,2,4,2,6,8,5,6,3,4,5,2,4,

%T 3,3,8,5,8,6,4,3,10,6,6,5,1,7,4,4,6,9,6,9,5,4,6,10,3,7,7,6,3,8,13,5,8,

%U 3,9,11,4,8,6,8,11,11,11,12,13,12,10,6,7,7,4,16,10,8,9,4,6,14,11,7,4,13,10,13,8,10

%N a(n) = |{0 < k < n: k*n + pi(k*n) is prime}|, where pi(.) is given by A000720.

%C Conjecture: a(n) > 0 for all n > 5.

%C This implies that there are infinitely many positive integers m with m + pi(m) prime.

%H Zhi-Wei Sun, <a href="/A237712/b237712.txt">Table of n, a(n) for n = 1..2500</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(6) = 1 since 2*6 + pi(2*6) = 12 + 5 = 17 is prime.

%e a(47) = 1 since 21*47 + pi(21*47) = 987 + 166 = 1153 is prime.

%t p[n_]:=PrimeQ[n+PrimePi[n]]

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

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

%Y Cf. A000040, A000720, A237578.

%K nonn

%O 1,7

%A _Zhi-Wei Sun_, Feb 24 2014