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A072676 Numbers n for which the prime circle problem has a solution composed of disjoint subsets: the arrangement of numbers 1 through 2n 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. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..72.

Eric Weisstein's World of Mathematics, Prime Circle.

EXAMPLE

n=10 is on the list 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 n=1 and n=4.

MATHEMATICA

n=10; lst={}; i=0; found=False; While[i<n&&!found, i++; If[i==n, found=PrimeQ[4n-1]&&PrimeQ[2n+1], found=PrimeQ[2n+2i-1]&&PrimeQ[2n+2i+1]&&PrimeQ[2i+1]]]; If[found, lst=Flatten[Table[{2j-1, 2n-2(j-i)}, {j, i, n}]], Print["no solution using this method"]]; If[found, While[n=i-1; n>0, i=0; found=False; While[i<n&&!found, i++; found=PrimeQ[2n+2i-1]&&PrimeQ[2n+2i+1]]; If[found, lst=Flatten[Append[Table[{2j-1, 2n-2(j-i)}, {j, i, n}], lst]]]]]; lst

CROSSREFS

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

Sequence in context: A160543 A023810 A102800 * A235986 A080197 A115847

Adjacent sequences:  A072673 A072674 A072675 * A072677 A072678 A072679

KEYWORD

nice,nonn

AUTHOR

T. D. Noe, Jul 01 2002

STATUS

approved

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Last modified July 19 12:32 EDT 2019. Contains 325159 sequences. (Running on oeis4.)