A232443 Number of ways to write n = p + q - pi(q), where p and q are odd primes, and pi(q) is the number of primes not exceeding q.

0, 0, 0, 1, 1, 2, 1, 2, 2, 2, 1, 2, 3, 4, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 3, 3, 2, 2, 5, 4, 5, 5, 3, 3, 4, 4, 4, 5, 4, 2, 6, 5, 5, 5, 3, 4, 8, 5, 5, 6, 2, 4, 7, 6, 6, 5, 3, 5, 7, 6, 7, 6, 4, 6, 6, 5, 7, 5, 6, 6, 7, 7, 7, 6, 4, 5, 7, 8, 8, 7, 7, 7, 7, 8, 9, 5, 6, 9, 9, 7, 6, 7, 6, 7, 8, 3, 8, 9, 5

Offset: 1

Author

Zhi-Wei Sun, Nov 23 2013

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 3. Moreover, every n = 6, 7, ... can be written as p + q - pi(q) with p, p + 6 and q all prime.

(ii) For each integer n > 7, there is a prime p < n with n + p - pi(p) prime.

(iii) Any integer n > 4 not equal to 9 or 17 can be written as p + q + pi(q) with p and q both prime.

(iv) Each integer n > 7 can be written as p + q + pi(p) + pi(q) with p and q both prime.

Examples

			a(4) = 1 since 4 = 3 + 3 - pi(3) with 3 prime.
a(5) = 1 since 5 = 3 + 5 - pi(5) with 3 and 5 prime.
a(6) = 2 since 6 = 3 + 7 - pi(7) = 5 + 3 - pi(3) with 3, 5, 7 all prime.
a(7) = 1 since 7 = 5 + 5 - pi(5) with 5 prime.
a(11) = 1 since 11 = 5 + 11 - pi(11) with 5 and 11 both prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Crossrefs

Cf. A000040, A000720, A002375, A232398.

Programs

Mathematica

PQ[n_]:=PQ[n]=n>2&&PrimeQ[n]
a[n_]:=Sum[If[PQ[n-Prime[k]+k],1,0],{k,2,PrimePi[2n-2]}]
Table[a[n],{n,1,100}]