A024364 -id:A024364 - cp's OEIS frontend

A326309 Ordered perimeters p of primitive Pythagorean triangles having short leg in common with the long leg or hypotenuse of a primitive Pythagorean triangle with perimeter < p, and also having both the long leg and the hypotenuse in common with the short legs of primitive Pythagorean triangles with perimeter > p.

30, 84, 182, 234, 240, 260, 306, 312, 374, 462, 476, 510, 532, 546, 570, 650, 828, 840, 870, 900, 920, 966, 986, 1050, 1100, 1188, 1254, 1260, 1276, 1320, 1330, 1364, 1392, 1406, 1508, 1554, 1612, 1624, 1716, 1722, 1736, 1794, 1820, 1850, 1860, 1886, 2070, 2214, 2220

Offset: 1

Torlach Rush, Oct 17 2019

nonn

The short leg of a primitive Pythagorean triangle of perimeter a(n) is either the long leg or hypotenuse of a triangle whose perimeter is less than a(n).
The long leg and the hypotenuse of a triangle with perimeter a(n) are the short legs of triangles with perimeter greater than a(n).
This sequence is a subsequence of A024364. A subsequence of this sequence exists after applying the restrictions imposed by the sequence title to the sequence itself and begins a(2), a(3), a(9), a(11), ... . Applying the same restrictions on {a(2), a(3), a(9), a(11), ...} gives a sequence a(9), a(11), a(22), a(25), ... .
Question: Does recursive application of this sequence to A024364 terminate?

			30 is a term because 30 = 5+12+13 and 12 = 3+4+5 and 84 = 12+35+37 and 182 = 13+84+85.
84 is a term because 84 = 12+35+37 and 30 = 5+12+13 and 1260 = 35+612+613 and 1406 = 37+684+685.
182 is a term because 182 = 13+84+85 and 30 = 5+12+13 and 476 = 84+187+205 and 374 = 85+132+157.

Subsequence of A024364.

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

Original entry on oeis.org

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

Torlach Rush, Nov 12 2019

nonn

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.

			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).

Subset of A024364.

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