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.

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

A179682 Least integer, k, greater than n such that t(k)*t(n) form a perfect square; t(i) is the i-th triangular number (A000217).

Original entry on oeis.org

1, 8, 24, 48, 80, 120, 168, 224, 49, 360, 440, 528, 624, 728, 840, 960, 1088, 1224, 1368, 1520, 1680, 1848, 2024, 2208, 242, 2600, 2808, 3024, 3248, 3480, 3720, 3968, 4224, 4488, 4760, 5040, 5328, 5624, 5928, 6240, 6560, 6888, 7224, 7568, 7920, 8280, 8648
Offset: 0

Views

Author

Robert G. Wilson v, Jul 24 2010

Keywords

Comments

It appears that a(n) = A033996(n) for most n. - Robert Israel, Feb 15 2019

Links

Crossrefs

Cf. A000217, A033996, A175497, A306415.

Programs

  • Maple
    f:= proc(n) local F,t,p,k0,d,k,a,j;
  p:= max(map(t -> `if`(t[2]::odd, t[1],NULL), [op(ifactors(n)[2]),op(ifactors(n+1)[2])]));
  if n mod p = 0 then k0:= n+p-1; d:= 1;
    else  k0:= n+1; d:= p-1;
  fi;
  t:= n*(n+1)/4;
  for a from k0 by p do
    for k in [a, a+d] do
       if issqr(k*(k+1)*t) then return k fi
  od od
end proc:
f(0):= 1:
map(f, [$0..100]); # Robert Israel, Feb 15 2019
  • Mathematica
    f[n_] := Block[{k = n + 1, n2 = n (n + 1)/2}, While[ !IntegerQ@ Sqrt[n2*k (k + 1)/2], k++ ]; k]; Array[f, 47, 0]
  • Python
    from sympy.ntheory.primetest import is_square
def A179682(n):
    m = n*(n+1)>>1
    k = n+1
    while not is_square(m*k*(k+1)>>1):
        k += 1
    return k # Chai Wah Wu, Mar 13 2023

Formula

From Robert Israel, Feb 15 2019: (Start)
a(n) <= A033996(n).
If n = A033996(j) then a(n) <= A033996(a(j)).
If n = a(j) < A033996(j) then a(n) <= A033996(j).
(End)

Extensions

Incorrect empirical g.f. removed by Robert Israel, Feb 15 2019

A152005 Numbers whose square is the product of two distinct tetrahedral numbers A000292.

Original entry on oeis.org

2, 140, 280, 1092, 166460, 189070, 665840, 804540, 845460, 34250920, 38336088, 133784560, 138535992, 225792840, 4998790160, 6301258040, 7559616818, 8367691640, 39991371446, 104637102152, 227490888350, 1497809326860, 296523233581822
Offset: 1

Views

Author

Jonathan Vos Post, Nov 19 2008

Keywords

Comments

There may be values that are not given in the recurrence shown. This sequence is suggested by Ulas, p. 11, who supplied the recurrence.
a(24) > 3*10^14. - Donovan Johnson, Jan 11 2012

Examples

			From _R. J. Mathar_, Jan 22 2009: (Start)
2 is in the sequence because 2^2 = 4*1 = T(2)*T(1).
140 is in the sequence 140^2 = 560*35 = T(14)*T(5) = 19600*1 = T(48)*T(1).
280 is in the sequence because 280^2 = 19600*4 = T(48)*T(2).
1092 is in the sequence because 1092^2 = 3276*364 = T(26)*T(12). (End)

Links

Crossrefs

Cf. A000292, A175497 (products distinct triangular numbers).

Programs

  • Mathematica
    (* This program is not suitable to compute more than a dozen terms. *)
terms = 12; imin = 1; imax = 3000;
Union[Reap[Do[k2 = i(i+1)(i+2)/6 j(j+1)(j+2)/6; k = Sqrt[k2]; If[IntegerQ[k], Print[k]; Sow[k]], {i, imin, imax}, {j, i+1, imax}]][[2, 1]]][[1 ;; terms]] (* Jean-François Alcover, Oct 31 2018 *)

Formula

a(n) = T(i)*T(j) where T(k) = A000292(k) = C(k+2,3) = k*(k+1)*(k+2)/6.

Extensions

Sequence replaced by sequence with no intermediate terms missing by R. J. Mathar, Jan 22 2009
a(15)-a(18) from Donovan Johnson, Jan 24 2009
a(19)-a(23) from Donovan Johnson, Jan 11 2012
Showing 1-3 of 3 results.

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