A237705 Number of primes p < n with pi(n-p) prime, where pi(.) is given by A000720.

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

Offset: 1

Author

Zhi-Wei Sun, Feb 11 2014

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 4, and a(n) = 1 only for n = 5, 12, 26, 27, 32, 68.

(ii) For any integer n > 5, there is a prime p <= n with pi(n+p) prime.

(iii) If n > 32, then pi((n-p)^2) is prime for some prime p < n. Also, for each n > 6 there is an odd prime p < 2*n with pi((n - (p-1)/2)^2) prime.

(iv) Any integer n > 11 can be written as p + q with p and pi(q^2 + q + 1) both prime.

(v) Each integer n > 34 can be written as k + m with k and m positive integers such that pi(k^2) and pi(2*m^2) are both prime.

Examples

			a(5) = 1 since 2 and pi(5-2) = pi(3) = 2 are both prime.
a(12) = 1 since 7 and pi(12-7) = pi(5) = 3 are both prime.
a(15) = 2 since 3 and pi(15-3) = pi(12) = 5 are both prime, and 11 and pi(15-11) = pi(4) = 2 are both prime.
a(26) = 1 since 23 and pi(26-23) = 2 are both prime.
a(27) = 1 since 23 and pi(27-23) = 2 are both prime.
a(32) = 1 since 29 and pi(32-29) = 2 are both prime.
a(68) = 1 since 37 and pi(68-37) = pi(31) = 11 are both prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A000720, A237284, A237578, A237582.

Programs

Mathematica

q[n_]:=PrimeQ[PrimePi[n]]
a[n_]:=Sum[If[q[n-Prime[k]],1,0],{k,1,PrimePi[n-1]}]
Table[a[n],{n,1,70}]