This site is supported by donations to The OEIS Foundation.

Thanks to everyone who made a donation during our annual appeal!
To see the list of donors, or make a donation, see the OEIS Foundation home page.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A113689 Number of semiprimes in clumps of size >1 through n^2 in the semiprime spiral. 6
 0, 0, 2, 6, 9, 13, 17, 21, 23, 31, 37, 45, 54, 59, 72, 77, 83, 93, 104, 116, 125, 140, 150, 164, 180, 188, 203, 219, 236, 255, 272, 287, 301, 317, 334, 354, 378, 403, 419, 430, 450, 475, 498, 521, 542, 560, 588, 608, 626, 652, 677, 698 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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, A113689, gives an enumeration of the number of semiprimes in clumps of size >1 through n^2, not looking past the square boundary. A113688 gives isolated semiprimes in the semiprime spiral, namely those semiprimes none of whose adjacent integers in the spiral are semiprimes. REFERENCES S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 1998) 188; 30 (#4, 1999-2000), 246-250. LINKS M. Stein and S. M. Ulam, An Observation on the Distribution of Primes, Amer. Math. Monthly 74, 43-44, 1967. M. Stein and S. M. Ulam and M. B. Wells, A Visual Display of Some Properties of the Distribution of Primes, Amer. Math. Monthly 71, 516-520, 1964. Eric Weisstein's World of Mathematics, Prime Spiral. Eric Weisstein's World of Mathematics, Semiprime EXAMPLE a(3) = 2 because there is one visible clump through 3^2 = 9, {4,6}, which two semiprimes are diagonally connected. a(4) = 6 because there are 6 semiprimes in the 2 visible clumps through 4^2 = 16, {4, 6, 14, 15}, {9, 10}. a(5) = 9 because there are 9 semiprimes in the 3 visible clumps through 5^2 = 25, {4, 6, 14, 15}, {9, 10, 25}, {21, 22}. ...................... ... 17 16 15 14 13 ... ... 18  5  4  3 12 ... ... 19  6  1  2 11 ... ... 20  7  8  9 10 ... ... 21 22 23 24 25 ... ...................... CROSSREFS Cf. A001107, A001358, A002939, A002943, A004526, A005620, A007742, A033951-A033954, A033988, A033989-A033991, A033996, A063826, A113688. Sequence in context: A171639 A054770 A184745 * A190707 A020960 A076522 Adjacent sequences:  A113686 A113687 A113688 * A113690 A113691 A113692 KEYWORD easy,nonn AUTHOR Jonathan Vos Post, Nov 05 2005 EXTENSIONS Corrected and extended by Alois P. Heinz, Jan 02 2011 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified January 19 21:52 EST 2019. Contains 319310 sequences. (Running on oeis4.)