A329392 Ordered perimeters p of primitive Pythagorean triangles no side of which is squarefree.

286, 1026, 1702, 1798, 3286, 3920, 4508, 5368, 6042, 6450, 6466, 6552, 7686, 7938, 8520, 8964, 9900, 10044, 10296, 10324, 10494, 11988, 13206, 13612, 13786, 13806, 14058, 14606, 15004, 15912, 16692, 17316, 18382, 18748, 20002, 20328, 21054, 22042, 23074, 24402, 24926, 25500, 25872, 26378, 27104

Offset: 1

Author

Torlach Rush, Nov 12 2019

Keywords

• nonn

Comments

There are no perimeters p of primitive Pythagorean triangles all sides of which are squarefree. This is because one side is twice the product of two relatively prime numbers not both odd and therefore even.

Many terms of this sequence can be obtained by scaling (3,4,5) the sides of the smallest primitive Pythagorean triangle. For example, a(1) = (3*39) + (4*11) + (5*25).

a(6) is the first term of the sequence which cannot be obtained by scaling (3,4,5). In fact there is no primitive Pythagorean triangle smaller than a(6) that can be scaled to a(6) in the manner above, and in the context of this sequence a(6) can be thought of as "primitive".

a(514) = 310464 is the smallest perimeter corresponding to two triangles, namely (3^2*7^2*263, 2^6*11*89, 5^2*5273) and (2^6*3^2*251, 7^2*11*37, 5*17^2*101). - Giovanni Resta, Nov 15 2019

a(n) is the inner product of two vectors the components of which are relatively prime.

Examples

			286 is a term because 286 = (2*2*11) + (3*3*13) + (5*5*5).
1026 is a term because 1026 = (3*3*3*11) + (2*2*2*2*19) + (5*5*17).
1702 is a term because 1702 = (3*3*37) + (2*2*7*23) + (5*5*29).
		

Crossrefs

Subset of A024364.