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
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..1000
- Erich Friedman, What's Special About This Number? (See entry 7056.)
Programs
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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 -
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 *) -
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 -
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
Extensions
Corrected (missing terms inserted) by R. J. Mathar, Jun 04 2010
Comments