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A113688
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Isolated semiprimes in the semiprime spiral.
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9
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65, 74, 249, 295, 309, 355, 422, 511, 545, 667, 669, 758, 926, 943, 979, 998, 1099, 1167, 1186, 1322, 1457, 1469, 1561, 1585, 1658, 1711, 1774, 1779, 1835, 1891, 1959, 1961, 1963, 2021, 2038, 2066, 2155, 2186, 2191, 2206, 2271, 2329, 2342
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Write the integers 1, 2, 3, 4, ... in a counterclockwise square spiral. Analogous to Ulam's marking the primes in the spiral and discovering unexpectedly many connected diagonals, we construct a semiprime spiral by marking the 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. A113689 gives an enumeration of the number of semiprimes in clumps of size >1 through n^2.
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REFERENCES
| S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 1998) 188; 30 (#4, 1999-2000), 246-250.
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.
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LINKS
| Eric Weisstein's World of Mathematics, "Prime Spiral".
Eric Weisstein's World of Mathematics, "Semiprime.".
Alois P. Heinz, Plot of semiprime spiral, contains all semiprimes <= 10000, isolated semiprimes are red.
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EXAMPLE
| ......................
... 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
| Cf. A001107, A001358, A002939, A002943, A004526, A005620, A007742, A033951-A033954, A033988, A033989-A033991, A033996, A063826.
Sequence in context: A095535 A095523 A060877 * A159758 A056693 A164282
Adjacent sequences: A113685 A113686 A113687 * A113689 A113690 A113691
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KEYWORD
| easy,nonn
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AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 05 2005
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EXTENSIONS
| Corrected and extended by Alois P. Heinz (heinz(AT)hs-heilbronn.de), Jan 02 2011
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