cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Aug 07 2009

Keywords

Comments

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

Examples

			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, ...
  ...

Links

Crossrefs

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.

Programs

  • Maple
    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
  • Mathematica
    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 *)

Formula

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

Extensions

Edited by R. J. Mathar, Aug 01 2010
Example edited by Jean-François Alcover, Dec 15 2019

A100452 Triangle read by rows, based on array described below.

Original entry on oeis.org

1, 3, 4, 7, 8, 9, 13, 14, 15, 16, 19, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 39, 40, 42, 44, 45, 48, 49, 49, 50, 51, 52, 55, 60, 63, 64, 63, 64, 66, 68, 70, 72, 77, 80, 81, 79, 80, 81, 84, 85, 90, 91, 96, 99, 100, 91, 92, 93, 96, 100, 102, 105, 112, 117, 120, 121
Offset: 1

Views

Author

N. J. A. Sloane, Nov 22 2004

Keywords

Comments

The interesting property of this array is that the main diagonal gives A000960.

Examples

			Array begins:
  1 4 9 16 25 36 49 64 81 100 ...
    3 8 15 24 35 48 63 80  99 ...
      7 14 21 32 45 60 77  96 ...
        13 20 30 44 55 72  91 ...
           19 28 42 52 70  90 ...
and triangle begins:
   1
   3  4
   7  8  9
  13 14 15 16
  19 20 21 24 25
  27 28 30 32 35 36
  ...

Links

Crossrefs

Cf. A000960, A100453.
Column sums give A100454.
Row 1 = A000290, row 2 = A000290 - 1, row 3 = A100451.
See also A100461.

Programs

  • Magma
    function t(n,k) // t = A100452
  if k eq 1 then return n^2;
  else return (n-k+1)*Floor((t(n,k-1) -1)/(n-k+1));
  end if;
end function;
[t(n,n-k+1): k in [1..n], n in [1..15]]; // G. C. Greubel, Apr 07 2023
  • Mathematica
    max=11; a[1, n_]:= n^2;
a[m_, n_]/; 1, ]=0;
t= Table[a[m, n], {m,max}, {n,m,max}];
Flatten[Table[t[[m-n+1, n]], {m,max}, {n,m}]] (* Jean-François Alcover, Feb 21 2012 *)
  • SageMath
    def t(n, k): # t = A100452
    if (k==1): return n^2
    else: return (n-k+1)*((t(n, k-1) -1)//(n-k+1))
flatten([[t(n, n-k+1) for k in range(1,n+1)] for n in range(1, 16)]) # G. C. Greubel, Apr 07 2023

Formula

Form an array a(m,n) (n >= 1, 1 <= m <= n) by: a(1,n) = n^2 for all n; a(m+1,n) = (n-m)*floor( (a(m,n)-1)/(n-m) ) for 1 <= m <= n-1.

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

Views

Author

Omar E. Pol, Aug 08 2009

Keywords

Links

Crossrefs

Cf. A008578, A161344, A161345, A163280, A164000, A164004, A164006.
Cf. A028347.

Programs

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

Formula

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)

Extensions

Extended by R. J. Mathar, Aug 09 2009
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.