A072617 Number of essentially different ways of arranging numbers 1 through 2n around a circle so that the sum of each pair of adjacent numbers is prime, with the odd and even numbers in order in opposite directions.

1, 1, 1, 1, 1, 2, 1, 1, 1, 0, 0, 2, 1, 1, 2, 0, 0, 1, 1, 1, 3, 0, 0, 1, 0, 0, 2, 1, 1, 2, 0, 0, 2, 1, 1, 1, 0, 0, 3, 0, 0, 1, 0, 0, 3, 0, 0, 3, 1, 1, 1, 1, 1, 3, 0, 0, 0, 0, 0, 5, 0, 0, 3, 0, 0, 4, 1, 1, 4, 0, 0, 2, 1, 1, 2, 0, 0, 2, 0, 0, 4, 0, 0, 5, 0, 0, 4, 1, 1, 5, 0, 0, 3, 1, 1, 2, 1, 1, 4, 0

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. Finding these solutions is very fast because there are only n possible solutions to try. See A072616 for the case where only the odd numbers or only the even numbers are in order. Note that a(2)=1 because the two solutions are essentially the same. Solutions can be printed by removing comments from the Mathematica program.

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

			a(6) = 2 because there are two ways: {1,10,3,8,5,6,7,4,9,2,11,12} and {1,4,3,2,5,12,7,10,9,8,11,6}.
		

Links

• T. D. Noe, Table of n, a(n) for n=1..1000
• Eric Weisstein's World of Mathematics, Prime Circle.

Crossrefs

Cf. A051252, A072616, A072618.

Programs

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[j