A072676 Numbers k for which the prime circle problem has a solution composed of disjoint subsets: the arrangement of numbers 1 through 2k around a circle is such that the sum of each pair of adjacent numbers is prime, the odd numbers are in order and the even numbers are in groups of decreasing sequences.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 63, 64, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 78, 79, 81, 82, 83, 84

Offset: 1

Author

T. D. Noe, Jul 01 2002

Keywords

• nice
• nonn

Comments

This is a generalization of A072618. The integer k is in this sequence if either (a) 4k-1 and 2k+1 are prime, or (b) 2k+2i-1, 2k+2i+1 and 2i+1 are prime for some 0 < i < k. The Mathematica program computes a prime circle for such k. It is very easy to show that there are prime circles for large k, such as 10^10.

Examples

			k=10 is a term because one solution is {1, 2, 3, 8, 5, 6, 7, 4, 9, 20, 11, 18, 13, 16, 15, 14, 17, 12, 19, 10} and the even numbers are in three decreasing sequences {2}, {8, 6, 4} and {20, 18, 16, 14, 12, 10}. Note that this solution contains {1, 2} and {1, 2, 3, 8, 5, 6, 7, 4}, which are solutions for k=1 and k=4.
		

Links

• Eric Weisstein's World of Mathematics, Prime Circle.

Crossrefs

Cf. A051252, A072616, A072617, A072618, A072184.

Programs

Mathematica

n=10; lst={}; i=0; found=False; While[i0, i=0; found=False; While[i