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A208243
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Number of ways to write 2n-1 = p+q, where p is a prime, and both q and q+2 are practical numbers (A005153).
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20
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0, 0, 1, 2, 3, 2, 2, 2, 2, 3, 4, 4, 3, 2, 3, 3, 5, 6, 3, 3, 4, 4, 5, 7, 4, 3, 4, 2, 5, 7, 4, 4, 5, 4, 5, 7, 4, 5, 8, 2, 5, 7, 5, 5, 6, 6, 4, 7, 4, 5, 9, 3, 5, 9, 4, 6, 6, 5, 5, 7, 3, 3, 7, 3, 6, 8, 5, 4, 8, 4, 5, 8, 4, 4, 5, 3, 5, 8, 6, 3, 6, 4, 5, 12, 5, 5, 5, 3, 6, 8, 5, 4, 8, 4, 4, 8, 4, 6, 9, 5
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OFFSET
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1,4
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COMMENTS
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Conjecture: a(n)>0 for all n=3,4,...
The author has verified this for n up to 2*10^8. It is known that there are infinitely many practical numbers q with q+2 also practical.
Zhi-Wei Sun also made the following similar conjectures:
(1) Each odd number n>5 can be written as p+q with p and p+6 both prime and q practical. Also, any odd number n>3 not equal to 55 can be written as p+q with p and p+2 both prime and q practical.
(2) Each integer n>10 can be written as x+y (x,y>0) with 6x-1 and 6x+1 both prime, and y and y+6 both practical.
Also, any integer n>=6360 can be written as x+y (x,y>0) with 6x-1 and 6x+1 both prime, and y and y+2 both practical.
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LINKS
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EXAMPLE
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a(14)=2 since 2*14-1=27=11+16=23+4, where 11 and 23 are primes, 16,16+2,4,4+2 are practical numbers.
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MATHEMATICA
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f[n_]:=f[n]=FactorInteger[n]
Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
Con[n_]:=Con[n]=Sum[If[Part[Part[f[n], s+1], 1]<=DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]]+1, 0, 1], {s, 1, Length[f[n]]-1}]
pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
a[n_]:=a[n]=Sum[If[pr[2k]==True&&pr[2k+2]==True&&PrimeQ[2n-1-2k]==True, 1, 0], {k, 1, n-1}]
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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