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A187757
Number of ways to write n=x+y (x,y>0) with 6x-1, 6x+1, 6y+1 and 6y+5 all prime.
6
0, 1, 2, 3, 2, 2, 2, 4, 3, 2, 2, 3, 4, 4, 2, 3, 2, 6, 6, 5, 4, 2, 6, 5, 4, 4, 2, 6, 4, 4, 4, 3, 5, 7, 5, 5, 3, 4, 9, 5, 6, 4, 5, 6, 4, 5, 5, 6, 7, 6, 6, 3, 7, 7, 6, 6, 4, 6, 6, 5, 6, 4, 7, 6, 7, 2, 3, 7, 7, 7, 5, 3, 5, 5, 7, 8, 5, 8, 8, 4, 5, 4, 10, 10, 6, 6, 2, 9, 6, 9, 7, 1, 8, 4, 5, 7, 3, 9, 5, 3
OFFSET
1,3
COMMENTS
Conjecture: a(n)>0 for all n>1.
This has been verified for n up to 10^9. It implies that there are infinitely many twin primes and also infinitely many cousin primes, since the interval [m!+2,m!+m] of length m-2 contains no prime for any integer m>1.
EXAMPLE
a(92)=1 since 92=40+52 with 6*40-1, 6*40+1, 6*52+1 and 6*52+5 all prime.
MATHEMATICA
a[n_]:=a[n]=Sum[If[PrimeQ[6k-1]==True&&PrimeQ[6k+1]==True&&PrimeQ[6(n-k)+1]==True&&PrimeQ[6(n-k)+5]==True, 1, 0], {k, 1, n-1}]
Do[Print[n, " ", a[n]], {n, 1, 100}]
KEYWORD
nonn,nice
AUTHOR
Zhi-Wei Sun, Jan 03 2013
STATUS
approved