Search: id:A072676
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%I A072676
%S A072676 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,
%T A072676 29,30,31,33,34,35,36,37,39,40,41,42,43,45,46,48,49,50,51,52,53,54,55,
%U A072676 56,57,58,60,61,63,64,66,67,68,69,70,72,73,74,75,76,78,79,81,82,83,84
%N 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 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.
%C A072676 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.
%H A072676 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PrimeCircle.html">Link to a section of The World of Mathematics.</
               a>
%e A072676 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.
%t A072676 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
%Y A072676 Cf. A051252, A072616, A072617, A072618, A072184.
%Y A072676 Sequence in context: A160543 A023810 A102800 this_sequence A080197 A115847 
               A032966
%Y A072676 Adjacent sequences: A072673 A072674 A072675 this_sequence A072677 A072678 
               A072679
%K A072676 nice,nonn
%O A072676 1,2
%A A072676 T. D. Noe (noe(AT)sspectra.com), Jul 01 2002

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