%I A113688
%S A113688 65,74,249,295,309,355,422,511,545,667,669,721,723,749,758
%N A113688 Isolated semiprimes in the semiprime spiral.
%C A113688 Write the integers 1, 2, 3, 4, ... in a counterclockwise square spiral. 
               Analogous to Ulam coloring in the primes in the spiral and discovering 
               unexpectedly many connected diagonals, we construct a semiprime spiral 
               by coloring in all semiprimes (A001358). Each integer has 8 adjacent 
               integers in the spiral, horizontally, vertically and diagonally. 
               Curious extended clumps coagulate, slightly denser towards the origin, 
               of semiprimes connected by adjacency. This sequence gives isolated 
               semiprimes in the semiprime spiral, namely those semiprimes none 
               of whose adjacent integers in the spiral are semiprimes. A113688 
               gives an enumeration of the number of semiprimes in clumps of size 
               >1 through n^2.
%D A113688 Stein, M. and Ulam, S. M. "An Observation on the Distribution of Primes." 
               Amer. Math. Monthly 74, 43-44, 1967.
%D A113688 Stein, M. L.; Ulam, S. M.; and Wells, M. B. "A Visual Display of Some 
               Properties of the Distribution of Primes." Amer. Math. Monthly 71, 
               516-520, 1964.
%D A113688 S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 
               1998) 188; 30 (#4, 1999-2000), 246-250.
%H A113688 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PrimeSpiral.html">"Prime Spiral"</a>.
%H A113688 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Semiprime.html">"Semiprime."</a>.
%e A113688 ......................
%e A113688 ... 17 16 15 14 13 ...
%e A113688 ... 18 5 4 3 12 ...
%e A113688 ... 19 6 1 2 11 ...
%e A113688 ... 20 7 8 9 10 ...
%e A113688 ... 21 22 23 24 25 ...
%Y A113688 Cf. A001107, A001358, A002939, A002943, A004526, A005620, A007742, A033951-A033954, 
               A033988, A033989-A033991, A033996, A063826.
%Y A113688 Sequence in context: A095535 A095523 A060877 this_sequence A159758 A056693 
               A164282
%Y A113688 Adjacent sequences: A113685 A113686 A113687 this_sequence A113689 A113690 
               A113691
%K A113688 easy,nonn
%O A113688 1,1
%A A113688 Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 05 2005