A113519 -id:A113519 - cp's OEIS frontend

A056105 First spoke of a hexagonal spiral.

1, 2, 9, 22, 41, 66, 97, 134, 177, 226, 281, 342, 409, 482, 561, 646, 737, 834, 937, 1046, 1161, 1282, 1409, 1542, 1681, 1826, 1977, 2134, 2297, 2466, 2641, 2822, 3009, 3202, 3401, 3606, 3817, 4034, 4257, 4486, 4721, 4962, 5209, 5462, 5721, 5986, 6257

Offset: 0

Henry Bottomley, Jun 09 2000

Also the number of (not necessarily maximal) cliques in the n X n grid graph. - Eric W. Weisstein, Nov 29 2017

			The spiral begins:
                   49--48--47--46--45
                   /                 \
                 50  28--27--26--25  44
                 /   /             \   \
               51  29  13--12--11  24  43
               /   /   /         \   \   \
             52  30  14   4---3  10  23  42  67
             /   /   /   /     \   \   \   \   \
           53  31  15   5   1===2===9==22==41==66==>
             \   \   \   \         /   /   /   /
             54  32  16   6---7---8  21  40  65
               \   \   \             /   /   /
               55  33   17--18--19--20  39  64
                 \   \                 /   /
                 56  34--35--36--37--38  63
                   \                     /
                   57--58--59--60--61--62

G. C. Greubel, Table of n, a(n) for n = 0..5000
Henry Bottomley, Illustration of initial terms
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
Eric Weisstein's World of Mathematics, Clique
Eric Weisstein's World of Mathematics, Grid Graph
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A285792 (prime terms), A113519 (semiprime terms).
Other spokes: A003215, A056106, A056107, A056108, A056109.
Other spirals: A054552.
Cf. A000290, A000567, A008810, A033577.

List([0..50], n -> 3*n^2-2*n+1); # G. C. Greubel, Dec 02 2018

[3*n^2-2*n+1: n in [0..50]]; // Wesley Ivan Hurt, Jul 06 2014

A056105:=n->3*n^2 - 2*n + 1: seq(A056105(n), n=0..50); # Wesley Ivan Hurt, Jul 06 2014

LinearRecurrence[{3, -3, 1}, {1, 2, 9}, 50] (* Harvey P. Dale, Nov 02 2011 *)
Table[3 n^2 - 2 n + 1, {n, 0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)
CoefficientList[Series[(-1 + x - 6 x^2)/(-1 + x)^3, {x, 0, 20}], x] (* Eric W. Weisstein, Nov 29 2017 *)

a(n)=3*n^2-2*n+1 /* Michael Somos, Aug 03 2006 */

[3*n^2-2*n+1 for n in range(50)] # G. C. Greubel, Dec 02 2018

a(n) = 3*n^2 - 2*n + 1.
a(n) = a(n-1) + 6*n - 5.
a(n) = 2*a(n-1) - a(n-2) + 6.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = A056106(n) - n = A056107(n) - 2*n.
a(n) = A056108(n) - 3*n = A056109(n) - 4*n = A003215(n) - 5*n.
A008810(3*n-1) = A056109(-n) = a(n). - Michael Somos, Aug 03 2006
G.f.: (1-x+6*x^2)/(1-3*x+3*x^2-x^3). - Colin Barker, Jan 04 2012
From Robert G. Wilson v, Jul 05 2014: (Start)
Each of the 6 primary spokes or rays has a generating formula as stated here:
1st:  90 degrees A056105 3n^2 - 2n + 1
2nd:  30 degrees A056106 3n^2 -  n + 1
3rd: 330 degrees A056107 3n^2      + 1
4th: 270 degrees A056108 3n^2 +  n + 1
5th: 210 degrees A056109 3n^2 + 2n + 1
6th: 150 degrees A003215 3n^2 + 3n + 1
Each of the 6 secondary spokes or rays has a generating formula as stated here:
1st:  60 degrees 12n^2 - 27n + 16
2nd: 360 degrees 12n^2 - 25n + 14
3rd: 300 degrees 12n^2 - 23n + 12
4th: 240 degrees 12n^2 - 21n + 10
5th: 180 degrees 12n^2 - 19n +  8
6th: 120 degrees 12n^2 - 17n +  6 = A033577(n+1)
(End)
a(n) = 1 + A000567(n). - Omar E. Pol, Apr 26 2017
a(n) = A000290(n-1) + 2*A000290(n), n >= 1. - J. M. Bergot, Mar 03 2018
E.g.f.: (1 + x + 3*x^2)*exp(x). - G. C. Greubel, Dec 02 2018

A113524 Semiprimes in A056106.

Original entry on oeis.org

25, 141, 185, 235, 291, 753, 851, 955, 1565, 1851, 2495, 3235, 3641, 4295, 5251, 5765, 6031, 6865, 8061, 9353, 9691, 10741, 11103, 14215, 14631, 15481, 16355, 16801, 17711, 21085, 25855, 27553, 28131, 28715, 29305, 29901

Offset: 1

Jonathan Vos Post, Jan 12 2006

Intersection of A056106 and A001358.

			a(1) = 25 because A056106(3) = 25 = 5^2 is semiprime.
a(36) = 29901 because A056106(100) = 29901 = 3 * 9967 is semiprime.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001358, A056106.

Cf. A113519, A113525, A113527, A113528, A113530.

Select[Array[3 #^2 - # + 1 &, 100], PrimeOmega[#] == 2 &] (* Michael De Vlieger, Mar 17 2021 *)

A113530 Semiprimes in A003215.

Original entry on oeis.org

91, 169, 217, 469, 721, 817, 1027, 1141, 1261, 1387, 2611, 2977, 3781, 3997, 4681, 5677, 5941, 6487, 6769, 7651, 7957, 8587, 9577, 10981, 11347, 12481, 12871, 14077, 14491, 15769, 16207, 17557, 18019, 18961, 20419, 20917, 21421, 22969, 24031

Offset: 1

Jonathan Vos Post, Jan 12 2006

Intersection of A003215 and A001358.

			a(1) = 91 because A003215(5) = (5+1)^3 - 5^3 = 91 = 7 * 13 is semiprime.
a(7) = 121 because A003215(7) = (7+1)^3 - 7^3 = 169 = 13^2 is semiprime; the two prime factors need not be distinct.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001358, A003215.

Cf. A113519, A113524, A113525, A113527, A113528.

Select[Array[3 #^2 + 3 # + 1 &, 90], PrimeOmega[#] == 2 &] (* Michael De Vlieger, Mar 17 2021 *)

A113525 Semiprimes in A056107.

Original entry on oeis.org

4, 49, 301, 589, 973, 2353, 2701, 3073, 4333, 5293, 5809, 6349, 6913, 7501, 8749, 9409, 10801, 11533, 13069, 14701, 15553, 16429, 23233, 24301, 25393, 27649, 30001

Offset: 1

Jonathan Vos Post, Jan 12 2006

Intersection of A056107 and A001358.

			a(1) = 4 because A056107(1) = 3*1^2 + 1 = 4 = 2^2 is semiprime.
a(16) = 9409 because A056107(56) = 3*56^2 + 1 = 9409 = 97^2 is semiprime.
a(27) = 30001 because A056107(100) = 3*100^2 + 1 = 30001 = 19 * 1579 is semiprime.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001358, A056107.

Cf. A113519, A113524, A113527, A113528, A113530.

Select[Array[3 #^2 + 1 &, 100], PrimeOmega[#] == 2 &] (* Michael De Vlieger, Mar 17 2021 *)

A113527 Semiprimes in A056108.

Original entry on oeis.org

15, 115, 155, 201, 253, 445, 785, 1345, 2215, 3503, 3711, 4145, 4841, 5853, 6395, 7855, 9131, 12353, 13535, 14353, 16503, 18331, 19281, 20255, 20751, 21253, 21761, 23853, 24935, 26603, 29503, 30101

Offset: 1

Jonathan Vos Post, Jan 12 2006

Intersection of A056108 and A001358.

			a(1) = 15 because A056108(2) = 15 = 3 * 5 is semiprime.
a(2) = 115 because A056108(6) = 115 = 5 * 23 is semiprime.
a(32) = 30101 because A056108(100) = 30101 = 31 * 971 is semiprime.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001358, A056108.

Cf. A113519, A113524, A113525, A113528, A113530.

Select[Array[3 #^2 + # + 1 &, 100], PrimeOmega[#] == 2 &] (* Michael De Vlieger, Mar 17 2021 *)

{a(n)} = {3*n^2 + n + 1 iff semiprime}. {a(n)} = A056108 INTERSECT A001358.

A113528 Semiprimes in A056109.

Original entry on oeis.org

6, 34, 57, 86, 121, 209, 262, 321, 386, 706, 1241, 1366, 1497, 2582, 2761, 3334, 3746, 3961, 4881, 5377, 6166, 6722, 7009, 7601, 8857, 9862, 10562, 10921, 12417, 13201, 14422, 15697, 17026, 17481, 17942, 18409, 19361, 19846, 20337, 21337, 22361

Offset: 1

Jonathan Vos Post, Jan 12 2006

Intersection of A056109 and A001358.

			a(1) = 6 because A056109(1) = 3*1^2 + 2*1 + 1 = 6 = 2 * 3 is semiprime.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001358, A056109.

Cf. A113519, A113524, A113525, A113527, A113530.

Select[Array[3 #^2 + 2 # + 1 &, 87], PrimeOmega[#] == 2 &] (* Michael De Vlieger, Mar 17 2021 *)

A113653 Isolated semiprimes in the hexagonal spiral.

Original entry on oeis.org

6, 51, 69, 82, 91, 183, 194, 221, 249, 265, 287, 289, 309, 314, 319, 323, 355, 371, 403, 417, 437, 469, 478, 511, 517, 519, 533, 579, 589, 649, 681, 689, 731, 749, 758, 807, 838, 849, 926, 943, 951, 961, 965, 979, 1011, 1018, 1037, 1055, 1057, 1067, 1077, 1099, 1126, 1145, 1149, 1154, 1159

Offset: 1

Jonathan Vos Post, Jan 16 2006

nonn

Isolated semiprimes in the hexagonal spiral of A003215 and A001399, which is centered on 0. Of course such a spiral can be constructed beginning with any integer. Centering on 0 gives the interesting partition and multigraph equalities of A001399.
Integers in A001358 which are not adjacent in any of six directions to any other integer in A001358 when arranged in the hexagonal spiral.
An analog of  A113688 "Isolated semiprimes in the [square] spiral," and of the hexagonal prime spiral of [Abbott 2005; Weisstein, "Prime Spiral", MathWorld].
Unfortunately the original submission (which has been preserved as the "dead" sequence A335704) omitted the number 44 from the spiral, which has caused an enormous amount of trouble. - N. J. A. Sloane, Jun 27 2020

			The spiral begins:
                120-119-118-117-116-115-114
                 /                         \
              121  85--84--83-*82*-81--80 113
               /   /                     \   \
            122  86  56--55--54--53--52  79 112
             /   /   /                 \   \   \
          123  87  57  33--32--31--30 *51* 78 111
           /   /   /   /             \   \   \   \
        124  88  58  34  16--15--14  29  50  77 110
         /   /   /   /   /         \   \   \   \   \
      125  89  59  35  17   5---4  13  28  49  76 109
       /   /   /   /   /   /     \   \   \   \   \   \
    126  90  60  36  18  *6*  0   3  12  27  48  75 108
     /   /   /   /   /   /   /   /   /   /   /   /   /
  127 *91* 61  37  19   7   1---2  11  26  47  74 107 146
     \   \   \   \   \   \         /   /   /   /   /   /
    128  92  62  38  20   8---9--10  25  46  73 106 145
       \   \   \   \   \             /   /   /   /   /
      129  93  63  39  21--22--23--24  45  72 105 144
         \   \   \   \                 /   /   /   /
        130  94  64  40--41--42--43--44  71 104 143
           \   \   \                     /   /   /
          131  95  65--66--67--68-*69*-70 103 142
             \   \                         /   /
            132  96--97--98--99-100-101-102 141
               \                             /
              133-134-135-136-137-138-139-140

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

P. Abbott, Re: Hexagonal Spiral, (alt link), May 11, 2005
H. Bottomley, Spokes of a Hexagonal Spiral.
R. J. Mathar, Maple program for A113653
Eric Weisstein's World of Mathematics, Prime Spiral.

Cf. A001358, A001399, A113688.
For the sequence of isolated primes see A335916.
Related sequences:
A113519 Semiprimes in 1st spoke of a hexagonal spiral starting at 1 (A056105).
A113524 Semiprimes in 2nd spoke of a hexagonal spiral (A056106).
A113525 Semiprimes in 3rd spoke of a hexagonal spiral (A056107).
A113527 Semiprimes in 4th spoke of a hexagonal spiral (A056108).
A113528 Semiprimes in 5th spoke of a hexagonal spiral (A056109).
A113530 Semiprimes in 6th spoke of a hexagonal spiral (A003215).

Corrected and edited by N. J. A. Sloane, Jun 27 2020. Thanks to Jeffrey K. Aronson for pointing out the error in the original submission.

Terms a(4) onwards corrected by R. J. Mathar, Jun 29 2020

A335704 Erroneous version of A113653.

Original entry on oeis.org

6, 51, 55, 69, 82, 183, 194, 249, 259, 287, 309, 314, 319

Offset: 1

dead

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.

			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

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

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.

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

{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}.

A343423 Prime numbers p such that Euclidean distance from origin to p in hexagonal grid sets a new record. Number '1' is placed at the origin and '2' at (1, 0). Number 'm' (m >= 3) is placed by moving one unit forward in the direction from 'm-2' to 'm-1', if m - 1 is not a prime; otherwise, making 1/6 turn counterclockwise at 'm-1' followed by moving one unit forward.

Original entry on oeis.org

2, 3, 5, 7, 11, 29, 31, 59, 89, 127, 131, 157, 191, 193, 223, 227, 251, 257, 409, 521, 719, 757, 797, 809, 877, 881, 967, 971, 1009, 1013, 1049, 1087, 1091, 1117, 1123, 1277, 1301, 1361, 1409, 1423, 1447, 1451, 1523, 1531, 1657, 1693, 1697, 1699, 5273, 5323

Offset: 1

Ya-Ping Lu, Apr 15 2021

nonn

			Hexagonal grid with integers up to 85:
                         29<---28<---27<---26<-7,25<=6,24<==5/23
                        /                       /              \\
                      30                      8                4/22
                     /                       /                    \\
                  31,53<-52<---51<---50<--9,49<--48<---47         3,21
                  /  \                    /              \        /  \
                54    32                10               1,46--->2    20
               /        \              /                    \           \
           55,79<--78<-33,77<--76<-11,75<--74<---73          45          19
            //             \           \           \           \        /
        56,80               34          12          72          44    18
         //                   \           \           \        /  \  /
     57,81                     35          13--->14->15,71-->16-->17,43
      //                         \                    /           /
  58,82                           36                70          42
   //                               \              /           /
59,83                                37--->38->39,69-->40--->41
   \\                                           /
  60,84                                       68
     \\                                      /
    61,85--->62--->63--->64--->65--->66--->67
Prime number (p), square of the distance (s) from p to origin, and index (n) in the sequence for p up to 71 are:
p:  2  3  5  7  11  13  17  19  23  29  31  37  41  43  47  53   59   61  67  71
s:  1  3  7  9  13  13   9   7   7  37  43  31  19   9   1  43  109  109  43   7
n:  1  2  3  4   5  --  --  --  --   6   7  --  --  --  --  --    8   --  --  --

Cf. A113519, A257745, A309755, A335916.

from sympy import isprime
dx = [2, 1, -1, -2, -1, 1]; dy = [0, 1, 1, 0, -1, -1]
x = 0; y = 0; rec = 0; d = 0
for n in range(2, 10001):
    if isprime(n-1) == 1: d += 1; d %= 6
    x += dx[d]; y += dy[d]; s = x*x + 3*y*y
    if isprime(n) == 1 and s > rec: print(n); rec = s

cp's OEIS Frontend

A056105 First spoke of a hexagonal spiral.

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A113524 Semiprimes in A056106.

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A113530 Semiprimes in A003215.

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A113525 Semiprimes in A056107.

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A113527 Semiprimes in A056108.

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A113528 Semiprimes in A056109.

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A113653 Isolated semiprimes in the hexagonal spiral.

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A335704 Erroneous version of A113653.

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