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 1-5 of 5 results.

A053990 Numbers of the form x*(x + 1)*y*(y + 1) ("bipronics") where x and y are distinct.

Original entry on oeis.org

0, 12, 24, 40, 60, 72, 84, 112, 120, 144, 180, 220, 240, 252, 264, 312, 336, 360, 364, 420, 432, 480, 504, 540, 544, 600, 612, 660, 672, 684, 760, 792, 840, 864, 924, 936, 1012, 1080, 1092, 1104, 1120, 1200, 1260, 1300, 1320, 1404, 1440, 1512, 1584, 1624
Offset: 1

Views

Author

Stuart M. Ellerstein (ellerstein(AT)aol.com), Apr 04 2000

Keywords

Examples

			Taking x=1, y=2 gives 2 * 6 =12

Crossrefs

Cf. A054731, A054734, A072389.

Programs

  • Mathematica
    bipr[{a_,b_}]:=a(a+1)b(b+1); Take[Union[bipr/@Subsets[Range[0,40], {2}]],50] (* Harvey P. Dale, Jun 01 2012 *)

Extensions

More terms from James Sellers, Apr 22 2000

A169836 Perfect squares that are a product of two distinct triangular numbers.

Original entry on oeis.org

0, 36, 900, 1225, 7056, 32400, 41616, 44100, 88209, 108900, 298116, 705600, 1368900, 1413721, 1498176, 2924100, 5336100, 8643600, 8820900, 9217296, 10432900, 15210000, 24147396, 37088100, 48024900, 50893956, 50979600, 52490025, 55353600, 80568576
Offset: 1

Views

Author

R. J. Mathar, May 30 2010

Keywords

Comments

a(47) = 1728896400 is the product of two distinct triangular numbers in two different ways. 1728896400 = A000217(8) * A000217(9800) = A000217(27) * A000217(3024). - Donovan Johnson, Sep 01 2012

Examples

			Examples: 900=3*300. 7056 = 6*1176. 1368900 = 6*228150. 44100 = 36*1225.

Links

Crossrefs

Cf. A000217, A054731, A169835.

Programs

  • PARI
    istriangular(n)=issquare(8*n+1)
isok(n) = {if (issquare(n), d = divisors(n); fordiv(n, d, if (d > sqrtint(n), break); if ((d != n/d) && istriangular(d) && istriangular(n/d), return (1)););); return (0);} \\ Michel Marcus, Jul 24 2013
  • Python
    from itertools import count, islice, takewhile
from sympy import divisors
from sympy.ntheory.primetest import is_square
def A169836_gen(): # generator of terms
    return filter(lambda k:not k or any(map(lambda d: is_square((d<<3)+1) and is_square((k//d<<3)+1), takewhile(lambda d:d**2A169836_list = list(islice(A169836_gen(),20)) # Chai Wah Wu, Mar 13 2023

Formula

a(n) = (A175497(n))^2. [From R. J. Mathar, Jun 03 2010]

Extensions

More terms from R. J. Mathar, Jun 03 2010

A054734 Numbers of the form 2*x*(x + 1)*y*(y + 1) where x and y are distinct.

Original entry on oeis.org

0, 24, 48, 80, 120, 144, 168, 224, 240, 288, 360, 440, 480, 504, 528, 624, 672, 720, 728, 840, 864, 960, 1008, 1080, 1088, 1200, 1224, 1320, 1344, 1368, 1520, 1584, 1680, 1728, 1848, 1872, 2024, 2160, 2184, 2208, 2240, 2400, 2520, 2600, 2640, 2808
Offset: 1

Views

Author

Stuart M. Ellerstein (ellerstein(AT)aol.com), Apr 22 2000

Keywords

Crossrefs

Cf. A053990, A054731.

Formula

a(n) = 2 * A053990(n). - Sean A. Irvine, Feb 20 2022

Extensions

More terms from James Sellers, Apr 22 2000

A334130 Numbers that can be written as a product of distinct triangular numbers.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 18, 21, 28, 30, 36, 45, 55, 60, 63, 66, 78, 84, 90, 91, 105, 108, 120, 126, 135, 136, 150, 153, 165, 168, 171, 180, 190, 198, 210, 216, 231, 234, 253, 270, 273, 276, 280, 300, 315, 325, 330, 351, 360, 378, 396, 406, 408, 420, 435, 450
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 14 2020

Keywords

Links

Crossrefs

Cf. A000217, A006472, A054731, A085780, A085782, A334129.

Programs

  • Maple
    N:= 1000: # for all terms <= N
S:= {0,1}:
for i from 2 do
  t:= i*(i+1)/2;
  if t > N then break fi;
  S:= S union select(`<=`,map(`*`,S,t),N)
od:
sort(convert(S,list)); # Robert Israel, Apr 21 2020

A134544 A051731 * A002260.

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 3, 4, 3, 4, 2, 2, 3, 4, 5, 4, 6, 6, 4, 5, 6, 2, 2, 3, 4, 5, 6, 7, 4, 6, 6, 8, 5, 6, 7, 8, 3, 4, 6, 4, 5, 6, 7, 8, 9, 4, 6, 6, 8, 10, 6, 7, 8, 9, 10
Offset: 1

Views

Author

Gary W. Adamson, Oct 31 2007

Keywords

Comments

Row sums = A007437: (1, 4, 7, 14, 16, 3, 29, 50, ...).
Left border = A000010.

Examples

			First few rows of the triangle:
  1;
  2,  2;
  2,  2,  3;
  3,  4,  3,  4;
  2,  2,  3,  4,  5;
  4,  6,  6,  4,  5,  6;
  2,  2,  3,  4,  5,  6,  7;
  4,  6,  6,  8,  5,  6,  7,  8;
  3,  4,  6,  4,  5,  6,  7,  8,  9;
  4,  6,  6,  8, 10,  6,  7,  8,  9, 10;
  ...

Crossrefs

Cf. A054731, A007437, A000010.

Formula

A051731 * A002260 as infinite lower triangular matrices.
Showing 1-5 of 5 results.

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

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