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%I #5 Jun 27 2020 12:12:18
%S 6,51,55,69,82,183,194,249,259,287,309,314,319
%N Erroneous version of A113653.
%C This is the erroneous version of A113653 that was submitted to the OEIS by _Jonathan Vos Post_ on Jan 16 2006. Because 44 was omitted from the spiral, not only are the terms here incorrect, but a large number of other sequences will also need to be corrected. For this reason the whole of the original submission has been preserved here with a different A-number. - _N. J. A. Sloane_, Jun 27 2020
%C Isolated semiprimes in the hexagonal spiral, embedded in the triangular lattice, are the analogy to A113688 "Isolated semiprimes in the [square] spiral," as well as analogous in another way to the hexagonal prime spiral of [Abbott 2005; Weisstein, "Prime Spiral", MathWorld]. A113519 Semiprimes in first spoke of a hexagonal spiral (A056105). A113524 Semiprimes in second spoke of a hexagonal spiral (A056106). A113525 Semiprimes in third spoke of a hexagonal spiral (A056107). A113527 Semiprimes in fourth spoke of a hexagonal spiral (A056108). A113528 Semiprimes in fifth spoke of a hexagonal spiral (A056109). A113530 Semiprimes in sixth spoke of a hexagonal spiral (A003215). This is embedded in the hexagonal spiral of A003215 and A001399, which is centered on zero; of course such a spiral can be constructed beginning with any integer. Centering on zero gives the interesting partition and multigraph equalities of A001399.
%D Abbott, P. (Ed.). "Mathematica One-Liners: Spiral on an Integer Lattice." Mathematica J. 1, 39, 1990.
%H P. Abbott, <a href="http://forums.wolfram.com/mathgroup/archive/2005/May/msg00336.html">Re: Hexagonal Spiral</a>, <a href="http://groups-beta.google.com/group/comp.soft-sys.math.mathematica">(alt link)</a>, May 11, 2005
%H H. Bottomley, <a href="/A003215/a003215.gif">Spokes of a Hexagonal Spiral.</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeSpiral.html">Prime Spiral.</a>
%F {a(n)} = {integers in A001358 which are not adjacent in any of six directions to any other integers in A001358 when arranged as the hexagonal spiral of A003215}.
%e Copy this as proportionally spaced text, make semiprimes bold, draw boundaries around clumps of adjacent semiprimes. For example, there is a triangular clump of three semiprimes: {4, 14, 15}; a linear clump of three semiprimes {49, 77, 111}; a linear clump of two semiprimes {247, 305}; an irregular clump of seven {115, 155, 201, 202, 203, 253, 254}; a clump of eighteen whose least element is 33 and greatest is 206; and a long branching clump of sixteen whose least element is 9 and greatest is 129.
%e .................209.208.207.206.205.204.203.202.201
%e ................210.162.161.160.159.158.157.156.155.200
%e ..............211.163.121.120.119.118.117.116.115.154.199
%e ............212.164.122.86..85..84..83..82..81.114.153.198
%e ..........213.165.123.87..57..56..55..54..53..80.113.152.197
%e ........214.166.124.88..58..33..32..31..30..52..79.112.151.196
%e ......215.167.125.89..59..34..16..15..14..29..51..78.111.150.195
%e ....216.168.126.90..60..35..17..5...4...13..28..50..77.110.149.194
%e ..217.169.127.91..61..36..18..6...0...3...12..27..49..76.109.148.193
%e 218.170.128.92..62..37..19..7...1...2...11..26..48..75.108.147.192.243
%e ..219.171.129.93..63..38..20..8...9...10..25..47..74.107.146.191.242
%e ....220.172.130.94..64..39..21..22..23..24..46..73.106.145.190.241
%e ......221.173.131.95..65..40..41..42..43..45..72.105.144.189.240
%e ........222.174.132.96..66..67..68..69..70..71.104.143.188.239
%e ..........223.175.133.97..98..99.100.101.102.103.142.187.238
%e ............224.176.134.135.136.137.138.139.140.141.186.237
%e ..............225.177.178.179.180.181.182.183.184.185.236
%e ................226.227.228.229.230.231.232.233.234.235
%Y Cf. A001358, A001399, A003215, A056105-A056109, A113688, A113519, A113524, A113525, A113528, A113527, A113530, A113688.
%K dead
%O 1,1
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