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A072618
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Numbers n for which the prime circle problem has a simple solution: the arrangement of numbers 1 through 2n around a circle is such that the sum of each pair of adjacent numbers is prime and the odd and even numbers are in order in opposite directions.
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5
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1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 18, 19, 20, 21, 24, 27, 28, 29, 30, 33, 34, 35, 36, 39, 42, 45, 48, 49, 50, 51, 52, 53, 54, 60, 63, 66, 67, 68, 69, 72, 73, 74, 75, 78, 81, 84, 87, 88, 89, 90, 93, 94, 95, 96, 97, 98, 99, 102, 105, 108, 111, 112, 113, 114, 117, 118
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OFFSET
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1,2
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COMMENTS
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A very restricted form of the prime circle problem whose sequence is A051252. This sequence lists the n for which A072617(n) is positive. See A072616 for the case where only the odd numbers or only the even numbers are in order.
There is a provable solution for n when either (a) 2n+1 and 2n+3 are prime, (b) 2k+1, 2k+3, 2k+2n+1 and 2k+2n+3 are prime for some 0 < k < n-1, or (c) 2n-1, 2n+1 and 4n-1 are primes. Part (a) is due to Mike Hennebry. Note that cases (a) and (b) involve 3 sets of twin primes. For n > 3, due to the form of twin primes, it can be shown that (a) implies not (b) and not (c).
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LINKS
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EXAMPLE
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n=6 is on the list because the simple solution is {1, 10, 3, 8, 5, 6, 7, 4, 9, 2, 11, 12}.
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MATHEMATICA
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For[lst={}; n=1, n<=100, n++, oddTable=Append[Table[2i-1, {i, n}], 1]; evenTable=Table[2n+2-2i, {i, n}]; evenTable=Join[evenTable, evenTable]; For[cnt=0; i=1, i<=n, i++, j=0; allPrime=True; While[j<n&&allPrime, j++; allPrime= PrimeQ[oddTable[[j]]+evenTable[[i+j-1]]]&& PrimeQ[oddTable[[j+1]]+evenTable[[i+j-1]]]]; If[allPrime, cnt++ ]]; If[cnt>0, AppendTo[lst, n]]]; lst
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PROG
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(Haskell)
import Data.List (transpose)
a072618 n = a072618_list !! (n-1)
a072618_list = filter f [1 ..] where
f x = any (all ((== 1) . a010051' . fromIntegral)) $
map cs [concat $ transpose [[2*x, 2*x-2 .. 2] , us] |
us <- map (uncurry (++) . (uncurry $ flip (, ))
. flip splitAt [1, 3 .. 2 * x]) [1 .. x]]
cs zs = (head zs + last zs) : zipWith (+) zs (tail zs)
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CROSSREFS
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KEYWORD
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nice,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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