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

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

Author

R. J. Mathar, May 30 2010

Keywords

• nonn

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

• Donovan Johnson, Table of n, a(n) for n = 1..1000
• Erich Friedman, What's Special About This Number? (See entry 7056.)

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