OFFSET
1,14
COMMENTS
Conjecture: (i) a(n) > 0 for all n > 12.
(ii) For any integer n > 2, there is a prime p < n such that r = |{q <= n-p: q and q + 2 are both prime}| is a square.
See also A237815 for a similar conjecture involving Sophie Germain primes.
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(13) = 1 since {q <= 13 - 2: q and q + 2 are both prime} = {3, 5, 11} has cardinality 3, and {3, 3 + 2} is a twin prime pair.
MATHEMATICA
TQ[n_]:=PrimeQ[n]&&PrimeQ[n+2]
sum[n_]:=Sum[If[PrimeQ[Prime[k]+2], 1, 0], {k, 1, PrimePi[n]}]
a[n_]:=Sum[If[TQ[sum[n-Prime[k]]], 1, 0], {k, 1, PrimePi[n-1]}]
Table[a[n], {n, 1, 80}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 13 2014
STATUS
approved