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A335704 Erroneous version of A113653. 2
6, 51, 55, 69, 82, 183, 194, 249, 259, 287, 309, 314, 319 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

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.

REFERENCES

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

LINKS

Table of n, a(n) for n=1..13.

P. Abbott, Re: Hexagonal Spiral, (alt link), May 11, 2005

H. Bottomley, Spokes of a Hexagonal Spiral.

Eric Weisstein's World of Mathematics, Prime Spiral.

FORMULA

{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}.

EXAMPLE

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

CROSSREFS

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

Sequence in context: A301436 A136383 A043084 * A292818 A173082 A113653

Adjacent sequences:  A335701 A335702 A335703 * A335705 A335706 A335707

KEYWORD

dead

STATUS

approved

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Last modified December 2 08:31 EST 2021. Contains 349437 sequences. (Running on oeis4.)