A295769 -id:A295769 - cp's OEIS frontend

A296097 Triangular numbers that can be represented as a product of two triangular numbers greater than 1, as a product of three triangular numbers greater than 1, and as a product of four triangular numbers greater than 1.

25200, 61425, 145530, 500500, 749700, 828828, 1185030, 2031120, 2162160, 2821500, 4573800, 5364450, 13857480, 17907120, 20991960, 21783300, 24643710, 27162135, 29610360, 30933045, 34283340, 37191000, 40901490, 60769800, 64292130, 71216145, 76576500, 90041490

Offset: 1

Alex Ratushnyak, Dec 04 2017

nonn

Triangular numbers in the products need not be distinct, that is, duplicates in the products are allowed.

A subsequence of A188630 and of A295769.

			25200 = 210 * 120 = 120 * 21 * 10 = 28 * 15 * 10 * 6.

Cf. A000217, A188630, A295769.

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.

Original entry on oeis.org

3, 36, 630, 25200, 25200, 2821500, 55030954895280, 55030954895280, 55030954895280, 55030954895280, 55030954895280, 55030954895280

Offset: 1

Alex Ratushnyak, Dec 04 2017

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)

			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)

Cf. A000217, A188630, A212616, A295769, A296097.

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

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

cp's OEIS Frontend

A296097 Triangular numbers that can be represented as a product of two triangular numbers greater than 1, as a product of three triangular numbers greater than 1, and as a product of four triangular numbers greater than 1.

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