A164004 -id:A164004 - cp's OEIS frontend

A163280 Square array read by antidiagonals where column k lists the numbers j whose largest divisor <= sqrt(j) is k.

1, 2, 4, 3, 6, 9, 5, 8, 12, 16, 7, 10, 15, 20, 25, 11, 14, 18, 24, 30, 36, 13, 22, 21, 28, 35, 42, 49, 17, 26, 27, 32, 40, 48, 56, 64, 19, 34, 33, 44, 45, 54, 63, 72, 81, 23, 38, 39, 52, 50, 60, 70, 80, 90, 100, 29, 46, 51, 68, 55, 66, 77, 88, 99, 110, 121, 31, 58, 57, 76, 65, 78, 84, 96, 108, 120, 132, 144

Offset: 1

Omar E. Pol, Aug 07 2009

This sequence is a permutation of the natural numbers A000027. Note that the first column is formed by 1 together with the prime numbers.

Column k contains exactly those numbers j=k*m where m is either a prime >= j or one of the numbers in row k of A163925. - Franklin T. Adams-Watters, Aug 12 2009

			Array begins:
   1,  4,  9,  16,  25,  36,  49,  64,  81, 100, 121, 144, ...
   2,  6, 12,  20,  30,  42,  56,  72,  90, 110, 132, 156, ...
   3,  8, 15,  24,  35,  48,  63,  80,  99, 120, 143, 168, ...
   5, 10, 18,  28,  40,  54,  70,  88, 108, 130, 154, 180, ...
   7, 14, 21,  32,  45,  60,  77,  96, 117, 140, 165, 192, ...
  11, 22, 27,  44,  50,  66,  84, 104, 126, 150, 176, 204, ...
  13, 26, 33,  52,  55,  78,  91, 112, 135, 160, 187, 216, ...
  17, 34, 39,  68,  65, 102,  98, 128, 153, 170, 198, 228, ...
  19, 38, 51,  76,  75, 114, 105, 136, 162, 190, 209, 264, ...
  23, 46, 57,  92,  85, 138, 119, 152, 171, 200, 220, 276, ...
  29, 58, 69, 116,  95, 174, 133, 184, 189, 230, 231, 348, ...
  31, 62, 87, 124, 115, 186, 147, 232, 207, 250, 242, 372, ...
  ...

Omar E. Pol, Illustration of initial terms of column 1: A008578
Omar E. Pol, Illustration of initial terms of column 2: A161344
Omar E. Pol, Illustration of initial terms of columns 1-4: A008578, A161344, A161345, A161424
Index entries for sequences that are permutations of the natural numbers

Rows 1-12: A000290, A002378, A005563, A164004, A100451, A164006, A164007, A164008, A164009, A164010, A164011, A164012.
Columns 1-12: A008578, A161344, A161345, A161424, A161835, A162526, A162527, A162528, A162529, A162530, A162531, A162532.
Another version: A163990.
Cf. A000027, A000040, A033676, A147861, A163100, A164000.

A163280 := proc(n,k) local r,T ; r := 0 ; for T from k^2 by k do if A033676(T) = k then r := r+1 ; if r = n then RETURN(T) ; fi; fi; od: end: # R. J. Mathar, Aug 09 2009

nmax = 12;
pm = Prime[nmax];
sDiv[n_] := Select[Divisors[n], #^2 <= n&][[-1]];
Clear[col]; col[k_] := col[k] = Select[Range[k pm], sDiv[#] == k&];
T[n_, k_ /; 1 <= k <= Length[col[k]]] := col[k][[n]];
Table[T[n-k+1, k], {n, 1, nmax}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 15 2019 *)

Column k lists the numbers j such that A033676(j)=k.

Edited by R. J. Mathar, Aug 01 2010

Example edited by Jean-François Alcover, Dec 15 2019

A163990 Square array read by antidiagonals where the row n lists the numbers k such that their largest divisor <= sqrt(k) equals n.

Original entry on oeis.org

1, 4, 2, 9, 6, 3, 16, 12, 8, 5, 25, 20, 15, 10, 7, 36, 30, 24, 18, 14, 11, 49, 42, 35, 28, 21, 22, 13, 64, 56, 48, 40, 32, 27, 26, 17, 81, 72, 63, 54, 45, 44, 33, 34, 19, 100, 90, 80, 70, 60, 50, 52, 39, 38, 23, 121, 110, 99, 88, 77, 66, 55, 68, 51, 46, 29, 144, 132, 120, 108

Offset: 1

Omar E. Pol, Aug 11 2009

This sequence is a permutation of the natural numbers.
Note that the first row is formed by 1 together the prime numbers and the first column are the squares of the natural numbers.
For more information see A163280, the main entry for this sequence. (See also A161344).

			Array begins:
1, 2, 3, 5, 7, 11,
4, 6, 8, 10, 14,
9, 12, 15, 18,
16, 20, 24,
25, 30,
36,
See also the array in A163280.

Index entries for sequences that are permutations of the natural numbers
Omar E. Pol, Illustration for A008578, the first row of the array [From _Omar E. Pol_, Oct 24 2009]
Omar E. Pol, Illustration for A161344, the second row of the array [From _Omar E. Pol_, Oct 24 2009]
Omar E. Pol, Illustration for A008578, A161344, A161345 and A161424 [From _Omar E. Pol_, Oct 24 2009]

Cf. A000027, A000040, A033676, A147861, A163100, A163280, A163281, A163991, A164000.
Cf. Rows: 1=A008578, 2=A161344, 3=A161345, 4=A161424, 5=A161835, 6=A162526, 7=A162527, 8=A162528, 9=A162529, 10=A162530, 11=A162531, 12=A162532.
Cf. Columns: 1=A000290, 2=A002378, 4=A164004, 6=A164006, 7=A164007, 8=A164008, 9=A164009, 10=A164010, 11=A164011, 12=A164012.

Row n lists the numbers k such that A033676(k)=n.

A217748 Number of regions with infinite area in the exterior of a regular n-gon with all diagonals drawn.

Original entry on oeis.org

1, 4, 10, 18, 28, 40, 54, 70, 88, 108, 130, 154, 180, 208, 238, 270, 304, 340, 378, 418, 460, 504, 550, 598, 648, 700, 754, 810, 868, 928, 990, 1054, 1120, 1188, 1258, 1330, 1404, 1480, 1558, 1638, 1720, 1804, 1890, 1978, 2068, 2160, 2254, 2350, 2448, 2548

Offset: 3

Martin Renner, Mar 23 2013

nonn

For n > 3 same as A028552(n-3).

			a(3) = 1 since the equilateral triangle has no diagonals and therefore one exterior region with infinite area.
a(4) = 4 since the two diagonals of the square divide the exterior in four regions with infinite area.
a(5) = 10 since the ten diagonals of the regular pentagon divide the exterior in ten regions with infinite area of two different shapes.

Cf. A004526, A007678, A028552, A164004, A217745, A217746.

a[n_] := n*(n - 3); a[3] = 1; Array[a, 50, 3] (* Amiram Eldar, Dec 10 2022 *)

a(n) = if(n == 3, 1, n*(n-3)); \\ Amiram Eldar, Dec 10 2022

a(n) = n*(n-3) for n > 3.
a(n) = A217745(n) - A217746(n).
From Amiram Eldar, Dec 10 2022: (Start)
Sum_{n>=3} 1/a(n) = 29/18.
Sum_{n>=3} (-1)^(n+1)/a(n) = 23/18 - 2*log(2)/3. (End)

A164005 Zero together with row 5 of the array in A163280.

Original entry on oeis.org

0, 7, 14, 21, 32, 45, 60, 77, 96, 117, 140, 165, 192, 221, 252, 285, 320, 357, 396, 437, 480, 525, 572, 621, 672, 725, 780, 837, 896, 957, 1020, 1085, 1152, 1221, 1292, 1365, 1440, 1517, 1596, 1677, 1760, 1845, 1932, 2021, 2112, 2205, 2300, 2397, 2496, 2597

Offset: 0

Omar E. Pol, Aug 08 2009

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3, -3, 1).

Cf. A008578, A161344, A161345, A163280, A164000, A164004, A164006.

Cf. A028347.

A033676 := proc(n) local a,d; a := 0 ; for d in numtheory[divisors](n) do if d^2 <= n then a := max(a,d) ; fi; od: a; end: A163280 := proc(n,k) local r,T ; r := 0 ; for T from k^2 by k do if A033676(T) = k then r := r+1 ; if r = n then RETURN(T) ; fi; fi; od: end: A164005 := proc(n) if n = 0 then 0; else A163280(5,n) ; fi; end: seq(A164005(n),n=0..80) ; # R. J. Mathar, Aug 09 2009

Join[{0, 7, 14}, Table[n*(n + 4), {n, 3, 50}]] (* G. C. Greubel, Aug 28 2017 *)

x='x+O('x^50); concat([0], Vec(x*(7 - 7*x + 4*x^3 - 2*x^4)/(1 - x)^3)) \\ G. C. Greubel, Aug 28 2017

Conjecture: a(n) = A100451(n+2). (See A163280.)
From G. C. Greubel, Aug 28 2017: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n >= 3.
a(n) = n*(n+4), n >= 3.
G.f.: x*(7 - 7*x + 4*x^3 - 2*x^4)/(1 - x)^3.
E.g.f.: x*(x+5)*exp(x) + 2*x + x^2. (End)

Extended by R. J. Mathar, Aug 09 2009

A256666 a(n) = ( 2n(2n^2 + 11n + 26) - (-1)^n + 1 )/16.

Original entry on oeis.org

0, 5, 14, 29, 51, 82, 123, 176, 242, 323, 420, 535, 669, 824, 1001, 1202, 1428, 1681, 1962, 2273, 2615, 2990, 3399, 3844, 4326, 4847, 5408, 6011, 6657, 7348, 8085, 8870, 9704, 10589, 11526, 12517, 13563, 14666, 15827, 17048, 18330, 19675, 21084, 22559

Offset: 0

Luce ETIENNE, Apr 07 2015

Consider a grid of small triangles of side 1 forming polygon with side n*(n+3): a(n) is the number of equilateral triangles of side length >=1 in this figure that are oriented with the sides of figure.
This sequence gives the number of triangles of all sizes in a ((n^2+3*n))-iamond with a 3*(2*n-1)-gon n>=1.
Equals (1/2)*Sum_{i=0..n-1} (n-i)*(n+1-i) + (-3 + (1/8)*Sum_{j=0..(2*n+3+(-1)^n)/4} (2*n+5-(-1)^n-4*j)*(2 n+5+(-1)^n-4*j) ) numbers of triangles in a direction and in the opposite direction.
It is also a way (3 stages) to surround triangular n^2-iamonds by 3*n triangles side 1: in first stage we obtain A045947, in second stage A248851, in third stage this sequence.

			From third comment: a(0)=0, a(1)=1+4, a(2)=4+10, a(3)=10+19, a(4)=20+31, a(5)=35+47, a(6)=56+67.

Colin Barker, Table of n, a(n) for n = 0..1000
Luce ETIENNE, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A002623, A028552, A045947, A117143, A164004, A248851.

[(4*n^3+22*n^2+52*n+1-(-1)^n)/16: n in [0..50]]; // Vincenzo Librandi, Apr 08 2015

Table[(4 n^3 + 22 n^2 + 52 n + 1 - (-1)^n)/16, {n, 0, 50}] (* Vincenzo Librandi, Apr 08 2015 *)
LinearRecurrence[{3,-2,-2,3,-1},{0,5,14,29,51},50] (* Harvey P. Dale, Aug 18 2020 *)

concat(0, Vec(x*(2*x^3-3*x^2-x+5)/((x-1)^4*(x+1)) + O(x^100))) \\ Colin Barker, Apr 07 2015

a(n) = 2*A248851(n) - A045947(n) + A004526(n+1).
G.f.: x*(2*x^3-3*x^2-x+5) / ((x-1)^4*(x+1)). - Colin Barker, Apr 07 2015
a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5) for n>4. - Colin Barker, Apr 07 2015

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A163280 Square array read by antidiagonals where column k lists the numbers j whose largest divisor <= sqrt(j) is k.

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A163990 Square array read by antidiagonals where the row n lists the numbers k such that their largest divisor <= sqrt(k) equals n.

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A217748 Number of regions with infinite area in the exterior of a regular n-gon with all diagonals drawn.

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A164005 Zero together with row 5 of the array in A163280.

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A256666 a(n) = ( 2*n*(2*n^2 + 11*n + 26) - (-1)^n + 1 )/16.

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A256666 a(n) = ( 2n(2n^2 + 11n + 26) - (-1)^n + 1 )/16.