%I #21 Mar 03 2014 13:54:46
%S 0,0,0,0,0,0,0,0,0,1,2,2,3,3,4,4,4,3,3,3,3,4,4,4,6,5,5,5,3,4,6,6,7,6,
%T 4,4,4,4,5,5,5,5,4,4,4,4,3,3,4,4,6,6,4,5,5,5,7,6,6,6,5,5,4,4,5,5,5,5,
%U 5,6,8,8,8,7,7,7,4,4,4,4
%N Number of primes p < n with q = floor((n-p)/4) and prime(q) - q + 1 both prime.
%C Conjecture: Let m > 0 and n > 2*m + 1 be integers. If m = 1 and 2 | n, or m = 3 and n is not congruent to 1 modulo 6, or m = 2, 4, 5, ..., then there is a prime p < n such that q = floor((n-p)/m) and prime(q) - q + 1 are both prime.
%C In the cases m = 1, 2, this gives refinements of Goldbach's conjecture and Lemoine's conjecture (see also A235189). For m > 2, the conjecture is completely new.
%C See also A238701 for a similar conjecture involving primes q with q^2 - 2 also prime.
%H Zhi-Wei Sun, <a href="/A238134/b238134.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014.
%e a(29) = 3 since 7, floor((29-7)/4) = 5 and prime(5) - 5 + 1 = 11 - 4 = 7 are all prime; 17, floor((29-17)/4) = 3 and prime(3) - 3 + 1 = 5 - 2 = 3 are all prime; 19, floor((29-19)/4) = 2 and prime(2) - 2 + 1 = 3 - 1 = 2 are all prime.
%t PQ[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]-n+1]
%t a[n_]:=Sum[If[PQ[Floor[(n-Prime[k])/4]],1,0],{k,1,PrimePi[n-1]}]
%t Table[a[n],{n,1,80}]
%Y Cf. A000040, A045917, A046927, A234694, A234695, A235189, A237705, A238701.
%K nonn
%O 1,11
%A _Zhi-Wei Sun_, Mar 03 2014