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A169836
Perfect squares that are a product of two distinct triangular numbers.
3
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
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
KEYWORD
nonn
AUTHOR
R. J. Mathar, May 30 2010
EXTENSIONS
More terms from R. J. Mathar, Jun 03 2010
STATUS
approved