A236541 Number of ways to write 2*n = k + m with 0 < k <= m such that prime(k) + m and k + prime(m) are both prime.

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

Offset

1,6

Comments

Conjecture: (i) a(n) > 0 for all n > 31.

(ii) Any positive even number can be written as k + m with k > 0 and m > 0 such that (prime(k) - k) + m and k + (prime(m) - m) are both prime.

(iii) Each integer n > 24 can be written as k + m with k > 0 and m > 0 such that prime(k) + m*(m-1) and k*(k-1) + prime(m) are both prime.

(iv) Any integer n > 15 can be written as k + m with k > 0 and m > 0 such that prime(k) + phi(m) and phi(k) + prime(m) are both prime.

Part (ii) of the conjecture in A232443 implies that any integer n > 7 can be written as k + m (k > 0, m > 0) with prime(k) + m = n + prime(k) - k prime.

Links

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

Example

a(9) = 1 since 2*9 = 8 + 10 with prime(8) + 10 = 19 + 10 = 29 and 8 + prime(10) = 8 + 29 = 37 both prime.
a(92) = 1 since 2*92 = 86 + 98 with prime(86) + 98 = 443 + 98 = 541 and 86 + prime(98) = 86 + 521 = 607 both prime.

Mathematica

p[k_, m_]:=PrimeQ[Prime[k]+m]
a[n_]:=Sum[If[p[k, 2n-k]&&p[2n-k, k], 1, 0], {k, 1, n}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000010, A000040, A232443, A232463.

Sequence in context: A260492 A308400 A369816 * A113206 A158800 A144024

Adjacent sequences: A236538 A236539 A236540 * A236542 A236543 A236544

Keyword

nonn

Author

Zhi-Wei Sun, Jan 28 2014

Status

approved