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 1-8 of 8 results.

A165412 Divisors of 2520.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 56, 60, 63, 70, 72, 84, 90, 105, 120, 126, 140, 168, 180, 210, 252, 280, 315, 360, 420, 504, 630, 840, 1260, 2520
Offset: 1

Views

Author

Reinhard Zumkeller, Sep 17 2009

Keywords

Comments

2520 is the largest and last of most highly composite numbers = A072938(7) = A002182(18) = 2520;
a(A000005(2520)) = a(48) = 2520 is the last term.
A242627(2520*n) = 9. - Reinhard Zumkeller, Jul 16 2014

Links

Crossrefs

Cf. A018336, A018357, A018388, A018412, A018444, A018490, A018559, and A018676 are subsets.
Cf. A018253, A018256, A018261, A018266, A018293, A018321, A018350, A018609, A178877, A178878, A178858, A178859, A178860, A178861, A178862, A178863, A178864.

Programs

A261144 Irregular triangle of numbers that are squarefree and smooth (row n contains squarefree p-smooth numbers, where p is the n-th prime).

Original entry on oeis.org

1, 2, 1, 2, 3, 6, 1, 2, 3, 5, 6, 10, 15, 30, 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210, 1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 21, 22, 30, 33, 35, 42, 55, 66, 70, 77, 105, 110, 154, 165, 210, 231, 330, 385, 462, 770, 1155, 2310, 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 21, 22, 26, 30, 33, 35, 39, 42
Offset: 1

Views

Author

Jean-François Alcover, Nov 26 2015

Keywords

Comments

If we define a triangle whose n-th row consists of all squarefree numbers whose prime factors are all less than prime(k), we get this same triangle except starting with a row {1}, with offset 1. - Gus Wiseman, Aug 24 2021

Examples

			Triangle begins:
1, 2;                        squarefree and 2-smooth
1, 2, 3, 6;                  squarefree and 3-smooth
1, 2, 3, 5, 6, 10, 15, 30;
1, 2, 3, 5, 6,  7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210;
...

Links

Crossrefs

Cf. A000079 (2-smooth), A003586 (3-smooth), A051037 (5-smooth), A002473 (7-smooth), A018336 (7-smooth & squarefree), A051038 (11-smooth), A087005 (11-smooth & squarefree), A080197 (13-smooth), A087006 (13-smooth & squarefree), A087007 (17-smooth & squarefree), A087008 (19-smooth & squarefree).
Row lengths are A000079.
Rightmost terms (or column k = 2^n) are A002110.
Rows are partial unions of rows of A019565.
Row n is A027750(A002110(n)), i.e., divisors of primorials.
Row sums are A054640.
Column k = 2^n-1 is A070826.
Multiplying row n by prime(n+1) gives A339195, row sums A339360.
A005117 lists squarefree numbers.
A056239 adds up prime indices, row sums of A112798.
A072047 counts prime factors of squarefree numbers.
A246867 groups squarefree numbers by Heinz weight, row sums A147655.
A329631 lists prime indices of squarefree numbers, sums A319246.
A339116 groups squarefree semiprimes by greater factor, sums A339194.
Cf. A000040, A001221, A006881, A071403, A209862, A319247.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, [1],
      sort(map(x-> [x, x*ithprime(n)][], b(n-1))))
    end:
T:= n-> b(n)[]:
seq(T(n), n=1..7);  # Alois P. Heinz, Nov 28 2015
  • Mathematica
    primorial[n_] := Times @@ Prime[Range[n]]; row[n_] := Select[ Divisors[ primorial[n]], SquareFreeQ]; Table[row[n], {n, 1, 10}] // Flatten

Formula

T(n-1,k) = A339195(n,k)/prime(n). - Gus Wiseman, Aug 24 2021

A087005 Divisors of 2310.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 21, 22, 30, 33, 35, 42, 55, 66, 70, 77, 105, 110, 154, 165, 210, 231, 330, 385, 462, 770, 1155, 2310
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 29 2003

Keywords

Comments

2310 = 11# = A002110(5);
divisors of 2310 are squarefree (A005117) and 11-smooth;
a(A000005(2310)) = a(32) = 2310 is the last term.

Links

Crossrefs

Cf. A018255, A018336, A087006, A087007, A087008.

Programs

  • Mathematica
    squarefrees[s_List] := Block[{a = Subsets[s]}, SortBy[Table[
Product[a[[n, m]], {m, 1, Length[a[[n]]]}], {n, 1, Length[a]}], Abs[#] &]]
A087005 = squarefrees[Prime[Range[5]]](* Fred Daniel Kline, Jan 25 2015 *)
Divisors[2310] (* Harvey P. Dale, Jun 27 2020 *)
  • PARI
    divisors(2310) \\ Charles R Greathouse IV, Jun 21 2017

A087006 Divisors of 30030.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 21, 22, 26, 30, 33, 35, 39, 42, 55, 65, 66, 70, 77, 78, 91, 105, 110, 130, 143, 154, 165, 182, 195, 210, 231, 273, 286, 330, 385, 390, 429, 455, 462, 546, 715, 770, 858, 910, 1001, 1155, 1365, 1430, 2002, 2145, 2310, 2730
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 29 2003

Keywords

Comments

30030 = 13# = A002110(6);
divisors of 30030 are squarefree (A005117) and 13-smooth;
a(A000005(30030)) = a(64) = 30030 is the last term.

Links

Crossrefs

Cf. A018255, A018336, A087005, A087007, A087008.

Programs

A087007 Divisors of 510510.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 21, 22, 26, 30, 33, 34, 35, 39, 42, 51, 55, 65, 66, 70, 77, 78, 85, 91, 102, 105, 110, 119, 130, 143, 154, 165, 170, 182, 187, 195, 210, 221, 231, 238, 255, 273, 286, 330, 357, 374, 385, 390, 429, 442, 455, 462, 510, 546, 561
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 29 2003

Keywords

Comments

510510 = 17# = A002110(7);
divisors of 510510 are squarefree (A005117) and 17-smooth;
a(A000005(510510)) = a(128) = 510510 is the last term.

Links

Crossrefs

Cf. A018255, A018336, A087005, A087006, A087008.

Programs

A087008 Divisors of 9699690.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 26, 30, 33, 34, 35, 38, 39, 42, 51, 55, 57, 65, 66, 70, 77, 78, 85, 91, 95, 102, 105, 110, 114, 119, 130, 133, 143, 154, 165, 170, 182, 187, 190, 195, 209, 210, 221, 231, 238, 247, 255, 266, 273, 285, 286, 323, 330
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 29 2003

Keywords

Comments

9699690 = 19# = A002110(8);
divisors of 9699690 are squarefree (A005117) and 19-smooth;
a(A000005(9699690)) = a(256) = 9699690 is the last term.

Links

Crossrefs

Cf. A018255, A018336, A087005, A087006, A087007.

Programs

A194359 Triangle of divisors of 210^n, each number occurring once.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210, 4, 9, 12, 18, 20, 25, 28, 36, 45, 49, 50, 60, 63, 75, 84, 90, 98, 100, 126, 140, 147, 150, 175, 180, 196, 225, 245, 252, 294, 300, 315, 350, 420, 441, 450, 490, 525, 588, 630, 700, 735, 882, 900
Offset: 0

Views

Author

T. D. Noe, Aug 26 2011

Keywords

Comments

The length of row k is A005917, the rhombic dodecahedral numbers, (k+1)^4 - k^4. The triangle has rows beginning with 2^k and ending with 210^k.

Links

Crossrefs

Cf. A018336, A194356, A194357, A194358.

Programs

  • Mathematica
    Join[{{1}}, Table[Complement[Divisors[210^n], Divisors[210^(n-1)]], {n, 9}]]
Take[DeleteDuplicates[Flatten[Divisors/@(210^Range[5])]],100] (* Harvey P. Dale, Sep 03 2020 *)

A086297 Squarefree kernels of 7-smooth numbers.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 6, 14, 15, 2, 6, 10, 21, 6, 5, 3, 14, 30, 2, 35, 6, 10, 42, 15, 6, 7, 10, 6, 14, 30, 21, 2, 70, 6, 15, 10, 3, 42, 30, 6, 14, 10, 105, 6, 14, 30, 5, 42, 2, 15, 70, 6, 21, 30, 10, 6, 42, 35, 30, 21, 6, 14, 10, 210, 6, 14, 15, 30, 3, 35, 10, 42, 2
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 15 2003

Keywords

Comments

a(n) = A007947(A002473(n));
a(n) <= 210 = 7*5*3*2; range = A018336.

Links

Crossrefs

Cf. A002473, A007947, A018336.

Programs

  • Mathematica
    f[n_] := If[Max[(p = FactorInteger[n][[;; , 1]])] <= 7, Times @@ p, 0]; Select[f /@ Range[300], # > 0 &] (* Amiram Eldar, Sep 06 2020 *)
Showing 1-8 of 8 results.

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