
COMMENTS

From Jon E. Schoenfield, Apr 21 2018: (Start)
Of the 482 triangular numbers < 55030954895280 that can be represented as a product of seven triangular numbers greater than 1, the only one that can be represented as a product of two triangular numbers greater than 1 is 218434391280, which cannot be represented as a product of 3 triangular numbers greater than 1. Thus, a(n) >= 55030954895280 for all n >= 7.
However, 55030954895280 can be represented (see Example section) as a product of k triangular numbers greater than 1 for all k in 1,2,...,12 (but not for k=13), so a(7) = a(8) = ... = a(12) = 55030954895280 (and, for each n > 12, a(n) > 55030954895280, or a(n) = 1 if no such number exists).
If, for some integer N > 12, it could be proved that a(N) = 1, then it would also be established that a(n) = 1 for every n > N. (End)


EXAMPLE

25200 is the smallest triangular number representable as a product of 2, 3 and 4 triangular numbers, 25200 = 210 * 120 = 120 * 21 * 10 = 28 * 15 * 10 * 6.
Therefore a(4)=25200.
Also, 25200 = 28 * 10 * 10 * 3 * 3, and therefore a(5)=25200.
From Jon E. Schoenfield, Apr 21 2018: (Start)
Let f(k_1, k_2, ..., k_m) = Product_{j=1..m} A000217(k_j) = Product_{j=1..m} (k_j*(k_j + 1)/2). Then, since no smaller number can be represented as a product of k triangular numbers greater than 1 for all k in 1,2,...,7,
a(7) = 55030954895280
= f(10491039)
= f(2261, 6560)
= f(6, 493, 6560)
= f(28, 39, 81, 323)
= f(17, 18, 27, 40, 116)
= f(4, 8, 17, 28, 38, 81)
= f(2, 3, 17, 18, 26, 28, 40)
= f(2, 2, 2, 2, 2, 17, 144, 532)
= f(2, 2, 2, 2, 12, 17, 18, 28, 40)
= f(2, 2, 2, 2, 2, 2, 3, 3, 40, 2261)
= f(2, 2, 2, 2, 2, 2, 2, 2, 16, 29, 532)
= f(2, 2, 2, 2, 2, 2, 2, 2, 2, 7, 40, 493)
and a(7) = a(8) = a(9) = a(10) = a(11) = a(12).
(End)
