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Search: id:A113688
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| A113688 |
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Isolated semiprimes in the semiprime spiral. |
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+0
9
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| 65, 74, 249, 295, 309, 355, 422, 511, 545, 667, 669, 721, 723, 749, 758
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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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.
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REFERENCES
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Stein, M. and Ulam, S. M. "An Observation on the Distribution of Primes." Amer. Math. Monthly 74, 43-44, 1967.
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.
S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 1998) 188; 30 (#4, 1999-2000), 246-250.
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LINKS
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Eric Weisstein's World of Mathematics, "Prime Spiral".
Eric Weisstein's World of Mathematics, "Semiprime.".
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EXAMPLE
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......................
... 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 ...
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CROSSREFS
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Cf. A001107, A001358, A002939, A002943, A004526, A005620, A007742, A033951-A033954, A033988, A033989-A033991, A033996, A063826.
Sequence in context: A095535 A095523 A060877 this_sequence A159758 A056693 A164282
Adjacent sequences: A113685 A113686 A113687 this_sequence A113689 A113690 A113691
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KEYWORD
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easy,nonn
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AUTHOR
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Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 05 2005
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