A169835 Perfect squares that are a product of two triangular numbers.

1, 9, 36, 100, 225, 441, 784, 900, 1225, 1296, 2025, 3025, 4356, 6084, 7056, 8281, 11025, 14400, 18496, 23409, 29241, 32400, 36100, 41616, 44100, 53361, 64009, 76176, 88209, 90000, 105625, 108900, 123201, 142884, 164836, 189225, 216225, 246016, 278784, 298116

Offset

1,2

Comments

Includes (except for 0) A000537 and 3/2*x*(x+1) for x in A132596. - Robert Israel, Jan 16 2015

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Erich Friedman, What's Special About This Number? (See entry 7056.)

Maple

N:= 10^6: # to get all terms <= N
A:= select(issqr, {seq(seq(a*(a+1)*b*(b+1)/4,
    b = a .. floor(sqrt(1/4+4*N/a/(a+1))-1/2)), a=1..floor(sqrt(4*N)))});
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(A, list)); # Robert Israel, Jan 16 2015

Mathematica

M = 10^6; (* to get all terms <= M *)
A = Union[Select[Flatten[Table[Table[(1/4) a (a+1) b (b+1), {b, a, Floor[ Sqrt[1/4 + 4M/(a (a+1))] - 1/2]}], {a, 1, Floor[Sqrt[4M]]}]], IntegerQ[ Sqrt[#]]&]] (* Jean-François Alcover, Mar 09 2019, after Robert Israel *)

Prog

(PARI) istriangular(n)=issquare(8*n+1) \\ now one can use ispolygonal(n, 3)
isok(n) = {if (issquare(n), fordiv(n, d, if (d > sqrtint(n), break); if (istriangular(d) && istriangular(n/d), return (1)); ); ); return (0); } \\ Michel Marcus, Jul 24 2013
(Haskell)
a169835 n = a169835_list !! (n-1)
a169835_list = f [] (tail a000217_list) (tail a000290_list) where
   f ts us'@(u:us) vs'@(v:vs)
     | u <= v = f (u : ts) us vs'
     | any p $ map (divMod v) ts = v : f ts us' vs
     | otherwise = f ts us' vs
     where p (q, r) = r == 0 && a010054 q == 1
-- Reinhard Zumkeller, Mar 03 2015
(Python)
from itertools import count, islice, takewhile
from sympy import divisors
from sympy.ntheory.primetest import is_square
def A169835_gen(): # generator of terms
    return filter(lambda k:any(map(lambda d: is_square((d<<3)+1) and is_square((k//d<<3)+1), takewhile(lambda d:d**2<=k, divisors(k)))), (m**2 for m in count(0)))
A169835_list = list(islice(A169835_gen(), 20)) # Chai Wah Wu, Mar 13 2023

Crossrefs

Superset of A000537. Cf. A000217, A132596, A169836.

Cf. A000290, A010054.

Sequence in context: A085037 A231678 A231682 * A231686 A231688 A000537

Adjacent sequences: A169832 A169833 A169834 * A169836 A169837 A169838

Keyword

nonn

Author

R. J. Mathar, May 30 2010

Extensions

Corrected (missing terms inserted) by R. J. Mathar, Jun 04 2010

Status

approved