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

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

1,1

Comments

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).

Links

Table of n, a(n) for n=1..46.

Example

(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.)

Prog

(PARI) 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)

Crossrefs

Cf. A259263, A111200.

Sequence in context: A195239 A122064 A320511 * A325608 A255103 A184542

Adjacent sequences: A259285 A259286 A259287 * A259289 A259290 A259291

Keyword

nonn

Author

Derek Orr, Jun 23 2015

Status

approved