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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.
2
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.
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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 28 2014
STATUS
approved