A237768 - OEIS

%I #10 Feb 13 2014 03:28:31

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

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

%U 1,3,3,4,4,3,3,4,4,4,4,1

%N Number of primes p < n with pi(n-p) a Sophie Germain prime, where pi(.) is given by A000720.

%C Conjecture: a(n) > 0 for all n > 4, and a(n) = 1 only for n = 5, 12, 20, 21, 26, 27, 30, 31, 32, 60, 61, 68, 69, 80, 81.

%C This is stronger than part (i) of the conjecture in A237705.

%C We have verified that a(n) > 0 for all n = 5, ..., 2*10^7.

%H Zhi-Wei Sun, <a href="/A237768/b237768.txt">Table of n, a(n) for n = 1..10000</a>

%e a(5) = 1 since 2, pi(5-2) = pi(3) = 2 and 2*2 + 1 = 5 are all prime.

%e a(12) = 1 since 7, pi(12-7) = pi(5) = 3 and 2*3 + 1 = 7 are all prime.

%e a(81) = 1 since 47, pi(81-47) = pi(34) = 11 and 2*11 + 1 = 23 are all prime.

%t sg[n_]:=PrimeQ[n]&&PrimeQ[2n+1]

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

%t Table[a[n],{n,1,80}]

%Y Cf. A000040, A000720, A005384, A237284, A237705, A237706.

%K nonn

%O 1,6

%A _Zhi-Wei Sun_, Feb 13 2014