

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
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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, 19992000), 246250.


LINKS

Table of n, a(n) for n=1..52.
M. Stein and S. M. Ulam, An Observation on the Distribution of Primes, Amer. Math. Monthly 74, 4344, 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, 516520, 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, A033951A033954, A033988, A033989A033991, 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



