OFFSET
1,12
COMMENTS
Conjecture: a(n) > 0 for all n > 9, and a(n) = 1 only for n = 8, 10, 11, 26, 33, 50, 52.
This implies that there are infinitely many positive integers n with pi(2*n) - pi(n) prime.
Recall that Bertrand's postulate proved by Chebyshev in 1850 asserts that pi(2*n) - pi(n) > 0 for all n > 0.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.
EXAMPLE
a(11) = 1 since 11 = 4 + 7 with pi(2*4) - pi(4) = 4 - 2 = 2 and pi(2*7) - pi(7) = 6 - 4 = 2 both prime.
a(26) = 1 since 26 = 13 + 13 with pi(2*13) - pi(13) = 9 - 6 = 3 prime.
a(33) = 1 since 33 = 6 + 27 with pi(2*6) - pi(6) = 5 - 3 = 2 and pi(2*27) - pi(27) = 16 - 9 = 7 both prime.
a(50) = 1 since 50 = 23 + 27 with pi(2*23) - pi(23) = 14 - 9 = 5 and pi(2*27) - pi(27) = 16 - 9 = 7 both prime.
a(52) = 1 since 52 = 21 + 31 with pi(2*21) - pi(21) = 13 - 8 = 5 and pi(2*31) - pi(31) = 18 - 11 = 7 both prime.
MATHEMATICA
p[n_]:=PrimeQ[PrimePi[2n]-PrimePi[n]]
a[n_]:=Sum[If[p[k]&&p[n-k], 1, 0], {k, 1, n/2}]
Table[a[n], {n, 1, 80}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Mar 20 2014
STATUS
approved