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

1,2

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

Links

Donovan Johnson, Table of n, a(n) for n = 1..1000

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

Formula

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

Example

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

Prog

(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**2<k, divisors(k)))), (m**2 for m in count(0)))
A169836_list = list(islice(A169836_gen(), 20)) # Chai Wah Wu, Mar 13 2023

Crossrefs

Cf. A000217, A054731, A169835.

Sequence in context: A166790 A001812 A229680 * A335220 A248108 A233003

Adjacent sequences: A169833 A169834 A169835 * A169837 A169838 A169839

Keyword

nonn

Author

R. J. Mathar, May 30 2010

Extensions

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

Status

approved