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

Author

Hugo Pfoertner, Jul 19 2003

Keywords

• nonn

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

Examples

			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.
		

Links

• 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

Crossrefs

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

Extensions

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

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