|
%I #20 Apr 23 2018 02:45:40
%S 3,36,630,25200,25200,2821500,55030954895280,55030954895280,
%T 55030954895280,55030954895280,55030954895280,55030954895280
%N 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.
%C From _Jon E. Schoenfield_, Apr 21 2018: (Start)
%C 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.
%C 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).
%C 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)
%F a(n) >= A212616(n) (unless a(n) = -1). - _Jon E. Schoenfield_, Apr 21 2018
%e 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.
%e Therefore a(4)=25200.
%e Also, 25200 = 28 * 10 * 10 * 3 * 3, and therefore a(5)=25200.
%e From _Jon E. Schoenfield_, Apr 21 2018: (Start)
%e 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,
%e a(7) = 55030954895280
%e = f(10491039)
%e = f(2261, 6560)
%e = f(6, 493, 6560)
%e = f(28, 39, 81, 323)
%e = f(17, 18, 27, 40, 116)
%e = f(4, 8, 17, 28, 38, 81)
%e = f(2, 3, 17, 18, 26, 28, 40)
%e = f(2, 2, 2, 2, 2, 17, 144, 532)
%e = f(2, 2, 2, 2, 12, 17, 18, 28, 40)
%e = f(2, 2, 2, 2, 2, 2, 3, 3, 40, 2261)
%e = f(2, 2, 2, 2, 2, 2, 2, 2, 16, 29, 532)
%e = f(2, 2, 2, 2, 2, 2, 2, 2, 2, 7, 40, 493)
%e and a(7) = a(8) = a(9) = a(10) = a(11) = a(12).
%e (End)
%Y Cf. A000217, A188630, A212616, A295769, A296097.
%K nonn,more
%O 1,1
%A _Alex Ratushnyak_, Dec 04 2017
%E a(7)-a(12) from _Jon E. Schoenfield_, Apr 21 2018
|
