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

1,6

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.

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.

Example

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.

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}]

Crossrefs

Cf. A000040, A000720, A002375, A232398.

Sequence in context: A213325 A043555 A242753 * A118821 A118824 A209402

Adjacent sequences: A232440 A232441 A232442 * A232444 A232445 A232446

Keyword

nonn

Author

Zhi-Wei Sun, Nov 23 2013

Status

approved