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A220091
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Number of ways to write n=p+q+(n mod 2)q with p>q and p, q, 6q-1, 6q+1 all prime
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1
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0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 2, 1, 2, 2, 2, 3, 2, 2, 2, 3, 1, 2, 1, 1, 1, 3, 2, 2, 3, 2, 1, 1, 2, 2, 1, 2, 2, 2, 3, 3, 3, 1, 2, 3, 2, 3, 3, 1, 1, 4, 2, 1, 3, 1, 1, 3, 4, 3, 3, 2, 1, 1, 3, 3, 1, 2, 2, 4, 4, 5, 3, 1, 1, 3, 2, 3, 3, 2, 2, 4, 2, 3, 3, 0, 1, 5, 2, 2, 3, 1, 0, 2, 3
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OFFSET
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1,11
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COMMENTS
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Conjecture: a(n)>0 for all even n>=8070 and odd n>=18680.
This conjecture unifies the twin prime conjecture, Goldbach's conjecture and Lemoine's conjecture. It has been verified for n up to 10^7.
Zhi-Wei Sun also made the following conjecture: Any integer n>=6782 can be written as p+q+(n mod 2)q with p>q and p, q, q-6, q+6 all prime, and any integer n>=4410 can be written as p+q+(n mod 2)q with p>q and p, q, 2q-3, 2q+3 all prime, and any integer n>=16140 can be written as p+q+(n mod 2)q with p>q and p, q, 3q-2, 3q+2 all prime.
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LINKS
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EXAMPLE
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a(31)=1 since 31=17+2*7 with 6*7-1 and 6*7+1 twin primes.
a(32)=1 since 32=29+3 with 6*3-1 and 6*3+1 twin primes.
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MATHEMATICA
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a[n_]:=a[n]=Sum[If[PrimeQ[6Prime[k]-1]==True&&PrimeQ[6Prime[k]+1]==True&&PrimeQ[n-(1+Mod[n, 2])Prime[k]]==True, 1, 0], {k, 1, PrimePi[(n-1)/(2+Mod[n, 2])]}]
Do[Print[n, " ", a[n]], {n, 1, 100}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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