A237769 - OEIS

%I #8 Feb 13 2014 04:58:12

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

%T 1,2,2,3,4,3,3,4,3,5,5,3,3,2,2,5,5,3,3,3,3,5,5,2,2,3,3,3,4,2,2,6,6,9,

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

%N Number of primes p < n with pi(n-p) - 1 and pi(n-p) + 1 both prime, where pi(.) is given by A000720.

%C Conjecture: (i) a(n) > 0 for all n > 8, and a(n) = 1 only for n = 9, 34, 35.

%C (ii) For any integer n > 4, there is a prime p < n such that 3*pi(n-p) - 1, 3*pi(n-p) + 1 and 3*pi(n-p) + 5 are all prime. Also, for each integer n > 8, there is a prime p < n such that 3*pi(n-p) - 1, 3*pi(n-p) + 1 and 3*pi(n-p) - 5 are all prime.

%C (iii) For any integer n > 6, there is a prime p < n such that phi(n-p) - 1 and phi(n-p) + 1 are both prime, where phi(.) is Euler's totient function.

%H Zhi-Wei Sun, <a href="/A237769/b237769.txt">Table of n, a(n) for n = 1..10000</a>

%e a(9) = 1 since 2, pi(9-2) - 1 = 3 and pi(9-2) + 1 = 5 are all prime.

%e a(34) = 1 since 19, pi(34-19) - 1 = pi(15) - 1 = 5 and pi(34-19) + 1 = pi(15) + 1 = 7 are all prime.

%e a(35) = 1 since 19, pi(35-19) - 1 = pi(16) - 1 = 5 and pi(35-19) + 1 = pi(16) + 1 = 7 are all prime.

%t TQ[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]

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

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

%Y Cf. A000010, A000040, A000720, A001359, A006512, A014574, A022004, A022005, A237705, A237706, A237768.

%K nonn

%O 1,10

%A _Zhi-Wei Sun_, Feb 13 2014