A072618 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.

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

Offset: 1

Author

T. D. Noe, Jun 25 2002

Keywords

• nice
• nonn

Comments

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).

Examples

			n=6 is on the list because the simple solution is {1, 10, 3, 8, 5, 6, 7, 4, 9, 2, 11, 12}.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Prime Circle.

Crossrefs

Cf. A051252, A072616, A072617.

Cf. A010051.

Programs

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)
-- Reinhard Zumkeller, Mar 17 2013

Mathematica

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[j0, AppendTo[lst, n]]]; lst

Extensions

More terms from Robert G. Wilson v, Jun 28 2002