A296098 a(n) is the smallest triangular number (A000217) that can be represented as a product of k triangular numbers greater than 1 for all k = 1,2,...,n. a(n)=-1 if no such triangular number exists.
3, 36, 630, 25200, 25200, 2821500, 55030954895280, 55030954895280, 55030954895280, 55030954895280, 55030954895280, 55030954895280
Offset: 1
Examples
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)
Formula
a(n) >= A212616(n) (unless a(n) = -1). - Jon E. Schoenfield, Apr 21 2018
Extensions
a(7)-a(12) from Jon E. Schoenfield, Apr 21 2018
Comments