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A072676
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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 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.
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2
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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; internal format)
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OFFSET
| 1,2
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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.
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LINKS
| Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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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.
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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
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CROSSREFS
| Cf. A051252, A072616, A072617, A072618, A072184.
Sequence in context: A160543 A023810 A102800 * A080197 A115847 A204315
Adjacent sequences: A072673 A072674 A072675 * A072677 A072678 A072679
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KEYWORD
| nice,nonn
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AUTHOR
| T. D. Noe (noe(AT)sspectra.com), Jul 01 2002
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