A086446 Integers representable as the product of the sum of three positive integers with the sum of their reciprocals: n=(x+y+z)*(1/x+1/y+1/z).

9, 10, 11, 14, 15, 18, 26, 30, 34, 35, 38, 42, 54, 55, 59, 62, 63, 70, 74, 82, 90, 95, 98, 102, 105, 122, 126, 131, 135, 138, 143, 158, 159, 170, 179, 190, 194, 195, 202, 203, 210, 215, 227, 230, 234, 238, 251, 255, 258, 266, 270, 278, 294, 297, 298, 310, 315

Offset

1,1

Comments

All terms of this sequence occur also in A085514. Bremner et al. have shown that the problem is equivalent to finding rational points of infinite order on the elliptic curve E_n : u^2 = v^3 + (n^2 - 6*n - 3)*v^2 + 16*n*v

The only values of n < 1000 with positive representations are shown in bold type in Table 1 in Section 8 of Bremner et al.'s paper (except for the singular value n=9 and the case n=10) - Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 09 2008

Links

Table of n, a(n) for n=1..57.

A. Bremner, R. K. Guy and R. Nowakowski, Which integers are representable as the product of the sum of three integers with the sum of their reciprocals?, Math. Comp. 61 (1993) 117-130.

A. MacLeod, The Knight's Problem

A. MacLeod, Elliptic Curves

Example

a(2)=(1+1+2)*(1/1+1/1+1/2)=10.
a(3)=(1+2+3)*(1/1+1/2+1/3)=6*(11/6)=11.
a(4)=(2+3+10)*(1/2+1/3+1/10)=14.
a(12)=(561+6450+13889)*(1/561+1/6450+1/13889)=42.

Crossrefs

Cf. A085514 (also negative x, y, z admitted).

Sequence in context: A134534 A125004 A085514 * A168042 A342147 A045522

Adjacent sequences: A086443 A086444 A086445 * A086447 A086448 A086449

Keyword

nonn

Author

Hugo Pfoertner, Jul 19 2003

Extensions

Corrected and extended by David J. Rusin, Jul 30 2003

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 09 2008

Status

approved