OFFSET

1,1

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)

FORMULA

a(n) >= A212616(n) (unless a(n) = -1). - Jon E. Schoenfield, Apr 21 2018

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)

CROSSREFS

KEYWORD

nonn,more

AUTHOR

Alex Ratushnyak, Dec 04 2017

EXTENSIONS

a(7)-a(12) from Jon E. Schoenfield, Apr 21 2018

STATUS

approved