OFFSET
1,7
COMMENTS
Conjecture: (i) a(n) > 0 for all n > 4.
(ii) Each integer n > 5 can be written as p + q (q > 0) with p and pi(2*q) - pi(q) both prime.
(iii) Any integer n > 2 not equal to 11 can be written as p + q with p prime and pi(2*q) - pi(q) a square.
Part (i) is a refinement of Goldbach's conjecture. It implies that there are infinitely many primes p with pi(2*p) - pi(p) prime.
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(5) = 1 since 2*5 = 7 + 3 with 7, 3 and pi(2*7) - pi(7) = 6 - 4 = 2 all prime.
a(6) = 1 since 2*6 = 7 + 5 with 7, 5 and pi(2*7) - pi(7) = 2 all prime.
a(11) = 1 since 2*11 = 11 + 11 with 11 and pi(2*11) - pi(11) = 8 - 5 = 3 both prime.
a(16) = 1 since 2*16 = 13 + 19 with 13, 19 and pi(2*13) - pi(13) = 9 - 6 = 3 all prime.
a(23) = 1 since 2*23 = 23 + 23 with 23 and pi(2*23) - pi(23) = 14 - 9 = 5 both prime.
a(44) = 1 since 2*44 = 59 + 29 with 59, 29 and pi(2*59) - pi(59) = 30 - 17 = 13 all prime.
a(166) = 1 since 2*166 = 103 + 229 with 103, 229 and pi(2*103) - pi(103) = 46 - 27 = 19 all prime.
MATHEMATICA
p[n_, k_]:=PrimeQ[PrimePi[2*Prime[k]]-k]&&PrimeQ[2n-Prime[k]]
a[n_]:=Sum[If[p[n, k], 1, 0], {k, 1, PrimePi[2n-1]}]
Table[a[n], {n, 1, 80}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Mar 22 2014
STATUS
approved