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A320447
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n such that all n - p are prime where p is prime in range n/2 <= p < n.
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5
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OFFSET
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1,1
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COMMENTS
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The following is a quotation from Hage-Hassan in his paper (see Link below): "The (concept of) right and left symmetry is fundamental in physics. This incites us to ask whether this symmetry is in (the) primes. Find the numbers n with a + a' = n. a, a' are primes and {a} are all the primes with: n/2 <= a < n and n = 2,3, ...".
By conjecture there are only 6 terms and this has been checked for n up to 10^8. Limiting the sequence to even numbers, by conjecture, the 4 numbers 10, 16, 36, 210 are the only even numbers that when partitioned to be the sum of two primes per the Goldbach conjecture use all primes p in the range n/2 <= p < n to provide solutions.
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LINKS
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EXAMPLE
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a(5)=36, because the primes in the range 18 <= p < 36 are {19, 23, 29, 31}. Also the complementary set {17, 13, 7, 5} has all its members prime. This is the 5th occurrence of such a number.
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MATHEMATICA
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plst[n_] := Select[Range[Ceiling[n/2], n-1], PrimeQ]; lst={}; Do[If[plst[n]!={}&&AllTrue[n-plst[n], PrimeQ], AppendTo[lst, n]], {n, 1, 1000}]; lst
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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