OFFSET
1,4
COMMENTS
Conjecture: (i) a(n) > 0 for all n > 2.
(ii) If n > 3 is neither 11 nor 125, then n can be written as k + m with k > 0 and m > 0 such that 6*k - 1, 6*k + 1, prime(m) + 2 and 3*prime(m) - 10 are all prime.
(iii) Any integer n > 458 can be written as p + q with q > 0 such that {p, p + 2} and {prime(q), prime(q) + 2} are both twin prime pairs.
This is much stronger than the twin prime conjecture. We have verified part (i) of the conjecture for n up to 2*10^7.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014
EXAMPLE
a(11) = 1 since {6*1 - 1, 6*1 + 1} = {5, 7} and {prime(10), prime(10) + 2} = {29, 31} are both twin prime pairs.
a(16) = 1 since {6*3 - 1, 6*3 + 1} = {17, 19} and {prime(13), prime(13) + 2} = {41, 43} are both twin prime pairs.
MATHEMATICA
p[n_]:=PrimeQ[6n-1]&&PrimeQ[6n+1]
q[n_]:=PrimeQ[Prime[n]+2]
a[n_]:=Sum[If[p[k]&&q[n-k], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 27 2014
STATUS
approved