1,1

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.

Abbott, P. (Ed.). "Mathematica One-Liners: Spiral on an Integer Lattice." Mathematica J. 1, 39, 1990.

P. Abbott, . "http://groups-beta.google.com/group/comp.soft-sys.math.mathematica", "Re: Hexagonal Spiral." posting. May 11, 2005.

H. Bottomley, Spokes of a Hexagonal Spiral.

Eric Weisstein's World of Mathematics, Prime Spiral.

{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}.

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.

.................209.208.207.206.205.204.203.202.201

................210.162.161.160.159.158.157.156.155.200

..............211.163.121.120.119.118.117.116.115.154.199

............212.164.122.86..85..84..83..82..81.114.153.198

..........213.165.123.87..57..56..55..54..53..80.113.152.197

........214.166.124.88..58..33..32..31..30..52..79.112.151.196

......215.167.125.89..59..34..16..15..14..29..51..78.111.150.195

....216.168.126.90..60..35..17..5...4...13..28..50..77.110.149.194

..217.169.127.91..61..36..18..6...0...3...12..27..49..76.109.148.193

218.170.128.92..62..37..19..7...1...2...11..26..48..75.108.147.192.243

..219.171.129.93..63..38..20..8...9...10..25..47..74.107.146.191.242

....220.172.130.94..64..39..21..22..23..24..46..73.106.145.190.241

......221.173.131.95..65..40..41..42..43..45..72.105.144.189.240

........222.174.132.96..66..67..68..69..70..71.104.143.188.239

..........223.175.133.97..98..99.100.101.102.103.142.187.238

............224.176.134.135.136.137.138.139.140.141.186.237

..............225.177.178.179.180.181.182.183.184.185.236

................226.227.228.229.230.231.232.233.234.235

Cf. A001358, A001399, A003215, A056105-A056109, A113688, A113519, A113524, A113525, A113528, A113527, A113530, A113688.

Sequence in context: A066303 A136383 A043084 this_sequence A133395 A050916 A011790

Adjacent sequences: A113650 A113651 A113652 this_sequence A113654 A113655 A113656

easy,nonn

Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 16 2006