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A220419
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Number of ways to write n=x+y (x>0, y>0) with 2x+1, 2y-1 and x^3+2y^3 all prime.
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6
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0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 3, 1, 1, 0, 2, 1, 2, 1, 2, 1, 2, 0, 1, 1, 0, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 4, 2, 1, 1, 1, 3, 2, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 3, 3, 1, 4, 1, 1, 0, 4, 2, 2, 3, 0, 1, 3, 2, 2, 1, 0, 5, 2, 0, 0, 1, 2, 2, 2, 2, 1, 2, 2, 3, 3, 2, 0, 1, 0, 2, 2, 4, 3, 2, 1, 3, 4, 2, 3
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OFFSET
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1,13
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COMMENTS
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Conjecture: a(n)>0 for all n>527.
This has been verified for n up to 2*10^7. It implies the Goldbach conjecture since 2(x+y)=(2x+1)+(2y-1).
Zhi-Wei Sun also made the following similar conjectures:
(1) Each integer n>1544 can be written as x+y (x>0, y>0) with 2x-1, 2y+1 and x^3+2y^3 all prime.
(2) Any odd number n>2060 can be written as 2p+q with p, q and p^3+2((q-1)/2)^3 all prime.
(3) Every integer n>25537 can be written as p+q (q>0) with p, p-6, p+6 and p^3+2q^3 all prime.
(4) Any even number n>1194 can be written as x+y (x>0, y>0) with x^3+2y^3 and 2x^3+y^3 both prime.
(5) Each integer n>3662 can be written as x+y (x>0, y>0) with 3(xy)^3-1 and 3(xy)^3+1 both prime.
(6) Any integer n>22 can be written as x+y (x>0, y>0) with (xy)^4+1 prime. Also, any integer n>7425 can be written as x+y (x>0, y>0) with 2(xy)^4-1 and 2(xy)^4+1 both prime.
(7) Every odd integer n>1 can be written as x+y (x>0, y>0) with x^4+y^2 prime. Moreover, any odd number n>15050 can be written as p+2q with p, q and p^4+(2q)^2 all prime.
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LINKS
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EXAMPLE
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a(25)=1 since 25=3+22 with 2*3+1, 2*22-1 and 3^3+2*22^3=21323 all prime.
a(26)=1 since 26=11+15 with 2*11+1, 2*15-1 and 11^3+2*15^3=8081 all prime.
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MATHEMATICA
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a[n_]:=a[n]=Sum[If[PrimeQ[2k+1]==True&&PrimeQ[2(n-k)-1]==True&&PrimeQ[k^3+2(n-k)^3]==True, 1, 0], {k, 1, n-1}]
Do[Print[n, " ", a[n]], {n, 1, 1000}]
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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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