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A238756
Number of ordered ways to write n = k + m (k > 0 and m > 0) such that 2*k + 1, prime(prime(k)) - prime(k) + 1 and prime(prime(m)) - prime(m) + 1 are all prime.
3
0, 1, 2, 3, 3, 2, 3, 3, 3, 4, 2, 5, 4, 3, 6, 4, 4, 3, 3, 6, 5, 5, 4, 6, 6, 5, 6, 2, 7, 5, 5, 6, 4, 4, 4, 5, 5, 8, 2, 5, 4, 5, 8, 2, 5, 2, 7, 4, 8, 6, 4, 5, 3, 8, 4, 7, 5, 3, 7, 7, 5, 7, 5, 7, 9, 8, 7, 5, 9, 7, 10, 9, 7, 7, 6, 9, 10, 4, 5, 5
OFFSET
1,3
COMMENTS
Conjecture: a(n) > 0 for all n > 1.
We have verified this for n up to 10^7.
The conjecture suggests that there are infinitely many primes p with 2*pi(p) + 1 and prime(p) - p + 1 both prime.
LINKS
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.
EXAMPLE
a(6) = 2 since 6 = 2 + 4 with 2*2 + 1 = 5, prime(prime(2)) - prime(2) + 1 = prime(3) - 3 + 1 = 3 and prime(prime(4)) - prime(4) + 1 = prime(7) - 7 + 1 = 17 - 6 = 11 all prime, and 6 = 3 + 3 with 2*3 + 1 = 7 and prime(prime(3)) - prime(3) + 1 = prime(5) - 5 + 1 = 11 - 4 = 7 both prime.
MATHEMATICA
p[k_]:=PrimeQ[Prime[Prime[k]]-Prime[k]+1]
a[n_]:=Sum[If[PrimeQ[2k+1]&&p[k]&&p[n-k], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 80}]
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Mar 05 2014
STATUS
approved