A056109 -id:A056109 - cp's OEIS frontend

A335704 Erroneous version of A113653.

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}.

A358205 a(n) is the least number k such that 1 + 2k + 3k^2 has exactly n prime divisors, counted with multiplicity.

Original entry on oeis.org

0, 2, 1, 13, 19, 7, 61, 331, 169, 1141, 6487, 898, 20581, 315826, 59947, 296143, 1890466, 6141994, 1359025, 49188715, 20490901, 264422320, 178328878, 1340590345, 9476420614, 5989636213, 72238539832, 103619599441, 668478672403, 794002910839, 417430195531

Offset: 0

Robert Israel, Nov 03 2022

nonn

a(n) is the least k such that A001222(A056109(k)) = n.

			a(5) = 7 because 1 + 2*7 + 3*7^2 = 162 = 2*3^4 has 5 prime divisors, counted with multiplicity.
From _Jon E. Schoenfield_, Nov 05 2022: (Start)
Let m = 1 + 2*k + 3*k^2. Since no such number m is divisible by 2^2, 5, or 7, the smallest number m having a given number of prime factors counted with multiplicity will tend to have a large number of 3's among its prime factors:
.
   n    k = a(n)                     m = 1 + 2*k + 3*k^2
  --  ------------  -----------------------------------------------------
   0             0                          1
   1             2                         17 (prime)
   2             1                          6 = 2 * 3
   3            13                        534 = 2 * 3    * 89
   4            19                       1122 = 2 * 3    * 11 * 17
   5             7                        162 = 2 * 3^4
   6            61                      11286 = 2 * 3^3  * 11 * 19
   7           331                     329346 = 2 * 3^4  * 19 * 107
   8           169                      86022 = 2 * 3^6  * 59
   9          1141                    3907926 = 2 * 3^5  * 11 * 17 * 43
  10          6487                  126256482 = 2 * 3^5  * 11^2 * 19 * 113
  11           898                    2421009 =     3^10 * 41
  12         20581                 1270773846 = 2 * 3^9  * 19 * 1699
  13        315826               299238818481 =     3^9  * 19 * 73 * ...
  14         59947                10781048322 = 2 * 3^10 * 11 * 43 * 193
  15        296143               263102621634 = 2 * 3^12 * 17 * 14561
  16       1890466             10721588872401 =     3^12 * 11 * 19 * ...
  17       6141994            113172283172097 =     3^16 * 2629057
  18       1359025              5540849569926 = 2 * 3^14 * 11^2 * 4787
  19      49188715           7258589148431106 = 2 * 3^17 * 28103531
  20      20490901           1259631112357206 = 2 * 3^15 * 17 * 73 * ...
  21     264422320         209757490471391841 =     3^16 * 11 * 17 * ...
  22     178328878          95403566542874409 =     3^19 * 19 * 83 * ...
  23    1340590345        5391547422002837766 = 2 * 3^19 * 11^2 * ...
  24    9476420614      269407642979285252217 =     3^22 * 2617 * ...
  25    5989636213      107627225904222216534 = 2 * 3^20 * 19 * 97 * ...
  26   72238539832    15655219911322828844337 =     3^22 * 11 * 19 * ...
  27  103619599441    32211064165147101736326 = 2 * 3^22 * 11 * 43 * ...
  28  668478672403  1340591206374369138728034 = 2 * 3^22 * 19 * 331 * ...
  29  794002910839  1891321867264002956873442 = 2 * 3^23 * 11 * 73 * ...
  30  417430195531   522743904423981537506946 = 2 * 3^25 * 11 * 17 * ...
.
As a result, the last digits of the ternary representation of a(n) tend to fall into a pattern:
.
   n      a(n)             a(n) in base 3
  --  ------------  ---------------------------
   0             0                          0_3
   1             2                          2_3
   2             1                          1_3
   3            13                        111_3
   4            19                        201_3
   5             7                         21_3
   6            61                       2021_3
   7           331                     110021_3
   8           169                      20021_3
   9          1141                    1120021_3
  10          6487                   22220021_3
  11           898                    1020021_3
  12         20581                 1001020021_3
  13        315826               121001020021_3
  14         59947                10001020021_3
  15        296143               120001020021_3
  16       1890466             10120001020021_3
  17       6141994            102120001020021_3
  18       1359025              2120001020021_3
  19      49188715          10102120001020021_3
  20      20490901           1102120001020021_3
  21     264422320         200102120001020021_3
  22     178328878         110102120001020021_3
  23    1340590345       10110102120001020021_3
  24    9476420614      220110102120001020021_3
  25    5989636213      120110102120001020021_3
  26   72238539832    20220110102120001020021_3
  27  103619599441   100220110102120001020021_3
  28  668478672403  2100220110102120001020021_3
  29  794002910839  2210220110102120001020021_3
  30  417430195531  1110220110102120001020021_3
(End)

Gerry Martens, Table of n, a(n) for n = 0..40

Cf. A001222, A056109, A086285, A122488.

N:= 18: # for a(0)..a(N)
V:= Array(0..N): count:= 0:
for k from 0 while count < N+1 do
  v:= numtheory:-bigomega(1+2*k+3*k^2);
if v <= N and V[v] = 0 then
    count:= count+1; V[v]:= k
fi
od:
convert(V,list);

a[n_] := Module[{i = 0},While[! PrimeOmega[1 + 2 i + 3 i^2] == n, i += 1]; i]
Table[a[n], {n, 0, 14}] (* Gerry Martens, Nov 05 2022 *)

a(21)-a(22) from Amiram Eldar, Nov 04 2022

a(23)-a(30) from Jon E. Schoenfield, Nov 05 2022

A066486 a(n) = min( x : x^3 + n^3 == 0 mod (x+n-1) ).

Original entry on oeis.org

1, 6, 17, 34, 57, 2, 121, 6, 23, 262, 321, 386, 55, 534, 617, 88, 3, 902, 61, 144, 77, 52, 9, 1634, 1777, 1926, 17, 2242, 2409, 344, 2761, 198, 3137, 4, 3537, 164, 535, 4182, 4409, 112, 93, 5126, 5377, 768, 413, 6166, 453, 920, 7009, 7302, 1043, 22, 8217, 224, 13, 9186, 5, 34, 10209, 188, 19, 1560, 11657

Offset: 1

Benoit Cloitre, Jan 02 2002

nonn

Cf. A066333.

a[n_] := For[x = 1, True, x++, If[Mod[x^3 + n^3, x + n - 1] == 0, Return[x]]]; Array[a, 24] (* Jean-François Alcover, Feb 17 2018 *)

a(n) = {my(k=1); while((k^3+n^3)%(k+n-1) != 0, k++); k; } \\ Altug Alkan, Feb 17 2018

a(n) = 3*n^2 - 4*n + 2 for n=1, 2, 3, 4, 5, 7, 10, 11, 12, 14, 15, 18, 24, 25, 26, 28, 29, 31, 33, 35, 38, ...

That is, in those cases a(n) = A056109(n-1). It appears that the corresponding indices are given by A133431 (i.e., 1 U A002504). - Michel Marcus, Feb 17 2018

More terms from Altug Alkan, Feb 17 2018

A113618 a(n) = 1 + 2n + 3n^2 + 4n^3 + 5n^4 + 6n^5 + 7n^6 + 8*n^7.

Original entry on oeis.org

1, 36, 1793, 24604, 167481, 756836, 2620201, 7526268, 18831569, 42374116, 87654321, 169343516, 309160393, 538155684, 899445401, 1451432956, 2271560481, 3460629668, 5147732449, 7495831836, 10708033241, 15034586596, 20780659593

Offset: 0

Jonathan Vos Post, Jan 14 2006

1 + 2x + 3x^2 + 4x^3 + 5x^4 + 6x^5 + 7*n^6 + 8*n^7 is the derivative of 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 = (x^9 - 1)/(x-1).

			1 + 2*8 + 3*8^2 + 4*8^3 + 5*8^4 + 6*8^5 + 7*8^6 + 8*8^7 = 18831569 = 173 * 199 * 547.
1 + 2*26 + 3*26^2 + 4*26^3 + 5*26^4 + 6*26^5 + 7*26^6 + 8*26^7 = 66490537361 is prime, the smallest prime in the sequence.

Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Cf. A000012, A005408, A056109, A056578, A056579.

[1+2*n+3*n^2+4*n^3+5*n^4+6*n^5+7*n^6+8*n^7: n in [1..43]] // Vincenzo Librandi, Dec 21 2010

Join[{1},Table[Total[Table[p*n^(p-1),{p,8}]],{n,30}]] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,36,1793,24604,167481,756836,2620201,7526268},30] (* Harvey P. Dale, Jul 16 2014 *)

G.f.: (1+28*x+1533*x^2+11212*x^3+18907*x^4+7956*x^5+679*x^6+4*x^7)/(x-1)^8. - R. J. Mathar, Dec 21 2010

a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8), with a(0)=1, a(1)=36, a(2)=1793, a(3)=24604, a(4)=167481, a(5)=756836, a(6)=2620201, a(7)=7526268. - Harvey P. Dale, Jul 16 2014

A131465 a(n)=4n^4-3n^3+2n^2-n+1.

Original entry on oeis.org

1, 3, 47, 259, 861, 2171, 4603, 8667, 14969, 24211, 37191, 54803, 78037, 107979, 145811, 192811, 250353, 319907, 403039, 501411, 616781, 751003, 906027, 1083899, 1286761, 1516851, 1776503, 2068147, 2394309, 2757611, 3160771

Offset: 0

Mohammad K. Azarian, Jul 26 2007

Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Cf. A084849, A056109, A056105.

Table[4n^4-3n^3+2n^2-n+1,{n,0,30}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,3,47,259,861},40] (* Harvey P. Dale, May 27 2017 *)

a(n)=4*n^4-3*n^3+2*n^2-n+1 \\ Charles R Greathouse IV, Oct 21 2022

A267942 Interleave (n-1)^2 + 2 and (n+1)^2 + 2.

Original entry on oeis.org

3, 3, 2, 6, 3, 11, 6, 18, 11, 27, 18, 38, 27, 51, 38, 66, 51, 83, 66, 102, 83, 123, 102, 146, 123, 171, 146, 198, 171, 227, 198, 258, 227, 291, 258, 326, 291, 363, 326, 402, 363, 443, 402, 486, 443, 531, 486, 578, 531, 627, 578, 678, 627, 731, 678, 786, 731

Offset: 0

Paul Curtz, Jan 22 2016

Trisections:
3,  6,  6,  27, 27, 66,  66, ... = 3*(1, 2, 2, 9, 9, 22, 22, ... ). See A056105.
3,  3,  18, 18, 51, 51, 102, ... = 3*(1, 1, 6, 6, 17, 17, ... ). See A056109.
2, 11,  11, 38, 38, 83,  83, ...   (== 2 (mod 9)).
The trisections also have the signature (1,2,-2,-1,1). The corresponding main sequence is 0, 0, 0, 0, 1, 1, 3, 3, ... = A161680(n) with each term duplicated.

			a(0) = (2+13)/5, a(1) = (13+2)/5, a(2) = (5+5)/5, a(3) = (29+1)/5, ... (using first formula).

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000290, A007395, A010701, A022998, A056105, A056109, A059100, A161680, A174474, A193356, A255844, A261327.

&cat [[(n-1)^2+2, (n+1)^2+2]: n in [0..50]]; // Vincenzo Librandi, Jan 23 2016

Flatten[Table[{n^2 - 2 n + 3, n^2 + 2 n + 3}, {n, 0, 30}]] (* Vincenzo Librandi, Jan 23 2016 *)
CoefficientList[Series[(3 - 7 x^2 + 4 x^3 + 2 x^4)/((1 - x)^3 (1 + x)^2), {x, 0, 56}], x] (* Michael De Vlieger, Jan 24 2016 *)

Vec((3-7*x^2+4*x^3+2*x^4)/((1-x)^3*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 22 2016

a(n) = (A261327(n+2) + A261327(n-3))/5.
a(n+1) = a(n) + (-1)^n * A022998(n), a(0)=3.
a(n+3) = a(n) + 3*A193356(n), a(0)=a(1)=3, a(2)=2.
a(n) = 3 + A174474(n).
a(2n) + a(2n+1) = A255844(n).
From Colin Barker, Jan 22 2016: (Start)
a(n) = (2*n^2 - 6*(-1)^n*n - 2*n + 3*(-1)^n + 21)/8.
a(n) = (n^2 - 4*n + 12)/4 for n even.
a(n) = (n^2 + 2*n + 9)/4 for n odd.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 4.
G.f.: (3 - 7*x^2 + 4*x^3 + 2*x^4) / ((1-x)^3*(1+x)^2).
(End)

More terms from Colin Barker, Jan 22 2016

A338369 Triangle read by rows: T(n,k) = (Sum_{i=0..n-k}(1+ki)^3)/(Sum_{i=0..n-k} (1+ki)) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 7, 1, 1, 10, 17, 13, 1, 1, 15, 31, 34, 21, 1, 1, 21, 49, 64, 57, 31, 1, 1, 28, 71, 103, 109, 86, 43, 1, 1, 36, 97, 151, 177, 166, 121, 57, 1, 1, 45, 127, 208, 261, 271, 235, 162, 73, 1, 1, 55, 161, 274, 361, 401, 385, 316, 209, 91, 1, 1, 66, 199, 349, 477, 556, 571, 519, 409, 262, 111, 1

Offset: 0

Werner Schulte, Nov 26 2020

Seen as a square array: (1) A(n,k) = T(n+k,k) = (k^2*n^2+k*(k+2)*n+2)/2 for n,k >= 0; (2) A(n,k) = A(n-1,k) + k*(1 + k*n) for k >= 0 and n > 0; (3) A(n,k) = A(n,k-1) + k*n*(n+1) - n*(n-1)/2 for n >= 0 and k > 0; (4) G.f. of row n >= 0: (2 + (n^2+3*n-4)*x + (n^2-n+2)*x^2) / (2*(1-x)^3).

			The triangle T(n,k) for 0 <= k <= n starts:
n \k :  0   1    2    3    4    5    6    7    8    9   10   11   12
====================================================================
   0 :  1
   1 :  1   1
   2 :  1   3    1
   3 :  1   6    7    1
   4 :  1  10   17   13    1
   5 :  1  15   31   34   21    1
   6 :  1  21   49   64   57   31    1
   7 :  1  28   71  103  109   86   43    1
   8 :  1  36   97  151  177  166  121   57    1
   9 :  1  45  127  208  261  271  235  162   73    1
  10 :  1  55  161  274  361  401  385  316  209   91    1
  11 :  1  66  199  349  477  556  571  519  409  262  111    1
  12 :  1  78  241  433  609  736  793  771  673  514  321  133    1
etc.

Cf. A000012 (column 0, main diagonal), A000217 (column 1), A056220 (column 2), A081271 (column 3), A118057 (column 4), A002061 (1st subdiagonal), A056109 (2nd subdiagonal), A085473 (3rd subdiagonal), A272039 (4th subdiagonal).

T[n_, k_] := Sum[(1 + k*i)^3, {i, 0, n - k}]/Sum[1 + k*i, {i, 0, n - k}]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Nov 26 2020 *)

for(n=0,12,for(k=0,n,print1((k^2*(n-k)^2+k*(k+2)*(n-k)+2)/2,", "));print(" "))

T(n,k) = (k^2*(n-k)^2 + k*(k+2)*(n-k) + 2)/2 for 0 <= k <= n.
T(n,0) = T(n,n) = 1 for n >= 0.
T(n,k) = T(n-1,k-1) + k*(n-k)*(n-k+1) - (n-k)*(n-k-1)/2 for 0 < k <= n.
T(n,k) = T(n-1,k) + k * (1+k*(n-k)) for 0 <= k < n.
G.f. of column k >= 0: (1 + (k^2+k-2)*t + (1-k)*t^2) * t^k / (1-t)^3.
E.g.f.: exp(x+y)*(2 + (x^2 + 2*x - 2)*y + (x^2 - 4*x + 2)*y^2 - (2*x - 5)*y^3 + y^4)/2. - Stefano Spezia, Nov 27 2020

cp's OEIS Frontend

A335704 Erroneous version of A113653.

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A358205 a(n) is the least number k such that 1 + 2*k + 3*k^2 has exactly n prime divisors, counted with multiplicity.

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A066486 a(n) = min( x : x^3 + n^3 == 0 mod (x+n-1) ).

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A113618 a(n) = 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6 + 8*n^7.

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Magma

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A131465 a(n)=4n^4-3n^3+2n^2-n+1.

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Mathematica

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A267942 Interleave (n-1)^2 + 2 and (n+1)^2 + 2.

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Magma

Mathematica

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Extensions

A338369 Triangle read by rows: T(n,k) = (Sum_{i=0..n-k}(1+k*i)^3)/(Sum_{i=0..n-k} (1+k*i)) for 0 <= k <= n.

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Formula

A358205 a(n) is the least number k such that 1 + 2k + 3k^2 has exactly n prime divisors, counted with multiplicity.

A113618 a(n) = 1 + 2n + 3n^2 + 4n^3 + 5n^4 + 6n^5 + 7n^6 + 8*n^7.

A338369 Triangle read by rows: T(n,k) = (Sum_{i=0..n-k}(1+ki)^3)/(Sum_{i=0..n-k} (1+ki)) for 0 <= k <= n.