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A218867
Number of prime pairs {p,q} with p>q and {p-4,q+4} also prime such that p+(1+(n mod 6))q=n if n is not congruent to 4 (mod 6), and p-q=n and q<n/2 if n=4 (mod 6).
10
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 2, 1, 2, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 0, 1, 2, 2, 2, 2, 0, 2, 2, 1, 1, 1, 2, 1, 0, 0, 1, 0, 2, 2, 0, 2, 1, 3, 0, 1, 1, 2, 2, 1, 0, 3, 2, 3, 0, 2, 1, 4, 1, 1, 2, 1, 3, 2
OFFSET
1,30
COMMENTS
Conjecture: a(n)>0 for all n>50000 with n different from 50627, 61127, 66503.
This conjecture implies that there are infinitely many cousin prime pairs. It is similar to the conjectures related to A219157 and A219055.
EXAMPLE
a(20)=1 since 20=11+3*3 with 11-4 and 3+4 prime. a(28)=1 since 28=41-13 with 41-4 and 13+4 prime.
MATHEMATICA
c[n_]:=c[n]=If[Mod[n+2, 6]==0, 1, -1-Mod[n, 6]]; d[n_]:=d[n]=2+If[Mod[n+2, 6]>0, Mod[n, 6], 0]; a[n_]:=a[n]=Sum[If[PrimeQ[Prime[k]+4] == True && PrimeQ[n+c[n]Prime[k]] == True && PrimeQ[n+c[n]Prime[k]-4]==True, 1, 0], {k, 1, PrimePi[(n-1)/d[n]]}]; Do[Print[n, " ", a[n]], {n, 100}]
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Nov 13 2012
STATUS
approved