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

1,6

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.

Links

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

Example

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.

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}]

Crossrefs

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

Sequence in context: A254296 A248371 A237768 * A340673 A031284 A064569

Adjacent sequences: A237702 A237703 A237704 * A237706 A237707 A237708

Keyword

nonn

Author

Zhi-Wei Sun, Feb 11 2014

Status

approved