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A199800
Number of ways to write n = p+q with p, 6q-1 and 6q+1 all prime
2
0, 0, 1, 2, 2, 2, 2, 3, 2, 3, 0, 4, 2, 4, 3, 2, 2, 3, 3, 5, 3, 3, 3, 4, 4, 3, 2, 4, 3, 5, 3, 4, 3, 5, 5, 6, 3, 4, 3, 5, 5, 5, 6, 5, 4, 5, 5, 6, 7, 5, 4, 5, 4, 7, 6, 4, 4, 4, 5, 6, 6, 5, 6, 7, 4, 5, 2, 4, 7, 5, 7, 4, 5, 6, 7, 7, 7, 5, 6, 4, 7, 4, 7, 7, 6, 5, 3, 5, 8, 7, 7, 5, 5, 6, 4, 5, 4, 5, 8, 7
OFFSET
1,4
COMMENTS
Conjecture: a(n)>0 for all n>11.
This implies the twin prime conjecture, and it has been verified for n up to 10^9.
Zhi-Wei Sun also made some similar conjectures, for example, any integer n>5 can be written as p+q with p, 2q-3 and 2q+3 all prime, and each integer n>4 can be written as p+q with p, 3q-2+(n mod 2) and 3q+2-(n mod 2) all prime.
EXAMPLE
a(3)=1 since 3=2+1 with 2, 6*1-1 and 6*1+1 all prime.
MATHEMATICA
a[n_]:=a[n]=Sum[If[PrimeQ[n-k]==True&&PrimeQ[6k-1]==True&&PrimeQ[6k+1]==True, 1, 0], {k, 1, n-1}]
Do[Print[n, " ", a[n]], {n, 1, 100}]
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Dec 21 2012
STATUS
approved