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A086446
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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).
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1
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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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
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REFERENCES
| 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.
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LINKS
| A. MacLeod, The Knight's Problem
A. MacLeod, Elliptic Curves
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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.
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CROSSREFS
| Cf. A085514 (also negative x, y, z admitted).
Sequence in context: A134534 A125004 A085514 * A168042 A045522 A054967
Adjacent sequences: A086443 A086444 A086445 * A086447 A086448 A086449
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KEYWORD
| nonn
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AUTHOR
| Hugo Pfoertner (hugo(AT)pfoertner.org), Jul 19 2003
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EXTENSIONS
| Corrected and extended by Dave Rusin (rusin(at)math.niu.edu), Jul 30 2003
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 09 2008
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