OFFSET
1,5
COMMENTS
Conjecture: (i) a(n) > 0 for all n > 2.
(ii) Any integer n > 4 can be written as p + q with q > 0 such that p and p - 1 + prime(q) are both prime.
(iii) Each integer n > 7 can be written as p + q with q > 0 such that prime(p) + sigma(q) is prime, where sigma(q) denotes the sum of all positive divisors of q.
Clearly, part (i) is stronger than Goldbach's conjecture.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
EXAMPLE
a(7) = 1 since 2*7 = 7 + 7 with 7 and prime(7) + 7 - 1 = 17 + 6 = 23 both prime.
a(14) = 1 since 2*14 = 11 + 17 with 11, 17 and prime(11) + 16 = 47 all prime.
a(92) = 1 since 2*92 = 47 + 137 with 47, 137 and prime(47) + 136 = 347 all prime.
MATHEMATICA
a[n_]:=Sum[If[PrimeQ[2n-Prime[k]]&&PrimeQ[Prime[Prime[k]]+2n-Prime[k]-1], 1, 0], {k, 1, PrimePi[2n-1]}]
Table[a[n], {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 04 2014
STATUS
approved