A111200 -id:A111200 - cp's OEIS frontend

A259263 Numbers of the form (m*k)^2/(m^2-k^2) for distinct integers m and k.

12, 18, 48, 72, 108, 147, 150, 162, 180, 192, 225, 240, 288, 300, 400, 405, 432, 448, 450, 578, 588, 600, 648, 720, 768, 882, 900, 960, 972, 980, 1008, 1100, 1152, 1200, 1260, 1323, 1350, 1452, 1458, 1600, 1620, 1728, 1792, 1800, 2025, 2028, 2100, 2160, 2178, 2312, 2352, 2400, 2592, 2700, 2880, 3042, 3072, 3150

Offset: 1

Derek Orr, Jun 22 2015

nonn

The odd numbers are much more rare than even numbers: 147, 225, 405, 1323, 2025, 3645, 3675, ... For 1 <= m <= 10^4 and 1 <= k <= m, there are 9217 total solutions. Of these solutions, only 679 are odd. See A259288.
Similarly, the reciprocals of these numbers can be represented as the difference in the reciprocals of two squares (i.e., there exists two distinct integers m and k satisfying 1/a(n) = 1/m^2 - 1/k^2).
If a(n) is a square, its square root is in A111200.

			(3*6)^2/(6^2-3^2) = 18^2/(3*9) = 12. So 12 is a member of this sequence.

Cf. A063664, A111200, A259288.

v=[];for(m=1,7500,for(n=1,m-1,if(type(s=(m*n)^2/(m^2-n^2))=="t_INT",v=concat(v,s))));vecsort(v,,8)

A259288 Odd numbers of the form (m*k)^2/(m^2-k^2) for distinct integers m and k.

Original entry on oeis.org

147, 225, 405, 1323, 2025, 3645, 3675, 4225, 5625, 7203, 7623, 10125, 11025, 11907, 14415, 17457, 17787, 18225, 18513, 19845, 24375, 24843, 27225, 30625, 32805, 33075, 38025, 42483, 49005, 50625, 53067, 61347, 64827, 65025, 68445, 68607, 77763, 81225, 91125, 91875, 98397, 99225, 105625, 107163, 117045, 119025

Offset: 1

Derek Orr, Jun 23 2015

nonn

The first term ending in a 9 seems to be 1225449, and the first term ending in a 1 is 136161.
For 1 <= m <= 10^4 and 1 <= k <= m, there are 9217 numbers of the form (m*k)^2/(m^2-k^2). Of these numbers, only 679 are odd.
If a(n) is not a square, then m = 9*k or m = 7*k. If a(n) is a square, m does not appear to be a multiple of k.
Let a(n) be a square generated by m_1 and k_1. If a(n-1) is generated by m_2 and k_2, then k_1 = k_2 and m_1 < m_2.
The reciprocals of these numbers can be represented as the difference in the reciprocals of two squares (i.e., there exists two distinct integers m and k satisfying 1/a(n) = 1/m^2 - 1/k^2).

			(84*12)^2/(84^2-12^2) = 84^2/48 = 147. So 147 is a member of this sequence. (Note that k=12 and m=84 and so m=7*k.)

Cf. A259263, A111200.

v=[]; for(m=1, 7500, for(n=1, m-1, if(type(s=(m*n)^2/(m^2-n^2))=="t_INT"&&(s%2), v=concat(v, s)))); vecsort(v, , 8)

cp's OEIS Frontend

A259263 Numbers of the form (m*k)^2/(m^2-k^2) for distinct integers m and k.

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