A237496 Number of ordered ways to write n = k + m (0 < k <= m) with pi(k) + pi(m) - 2 prime, where pi(.) is given by A000720.

0, 0, 0, 0, 0, 1, 2, 4, 4, 3, 2, 3, 3, 3, 5, 3, 1, 4, 5, 5, 7, 4, 1, 2, 1, 1, 1, 1, 1, 3, 6, 7, 8, 8, 8, 8, 8, 9, 11, 11, 11, 11, 9, 7, 7, 4, 1, 2, 1, 2, 3, 5, 7, 10, 14, 14, 14, 10, 6, 10, 14, 16, 19, 16, 13, 12, 11, 10, 7, 6, 5, 3, 3, 4, 3, 6, 9, 13, 17, 18

Offset

1,7

Comments

Conjecture: (i) a(n) > 0 for all n > 5.

(ii) Any integer n > 23 can be written as k + m (k > 0 and m > 0) with pi(k) + pi(m) prime. Also, each integer n > 25 can be written as k + m (k > 0 and m > 0) with pi(k) + pi(m) - 1 prime.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..3000

Example

a(6) = 1 since 6 = 3 + 3 with pi(3) + pi(3) - 2 = 2 + 2 - 2 = 2 prime.
a(17) = 1 since 17 = 2 + 15 with pi(2) + pi(15) - 2 = 1 + 6 - 2 = 5 prime.
a(99) = 1 since 99 = 1 + 98 with pi(1) + pi(98) - 2 = 0 + 25 - 2 = 23 prime.

Mathematica

PQ[n_]:=n>0&&PrimeQ[n]
p[k_, m_]:=PQ[PrimePi[k]+PrimePi[m]-2]
a[n_]:=Sum[If[p[k, n-k], 1, 0], {k, 1, n/2}]
Table[a[n], {n, 1, 80}]

Crossrefs

Cf. A000040, A000720, A232465, A237284, A237291, A237453, A237497.

Sequence in context: A008777 A306691 A354954 * A202690 A342754 A257978

Adjacent sequences: A237493 A237494 A237495 * A237497 A237498 A237499

Keyword

nonn

Author

Zhi-Wei Sun, Feb 08 2014

Status

approved