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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).

%I A086446 #14 Oct 12 2017 08:21:59
%S A086446 9,10,11,14,15,18,26,30,34,35,38,42,54,55,59,62,63,70,74,82,90,95,98,
%T A086446 102,105,122,126,131,135,138,143,158,159,170,179,190,194,195,202,203,
%U A086446 210,215,227,230,234,238,251,255,258,266,270,278,294,297,298,310,315
%N 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).
%C A086446 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
%C A086446 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
%H A086446 A. Bremner, R. K. Guy and R. Nowakowski, <a href="https://doi.org/10.1090/S0025-5718-1993-1189516-5">Which integers are representable as the product of the sum of three integers with the sum of their reciprocals?</a>, Math. Comp. 61 (1993) 117-130.
%H A086446 A. MacLeod, <a href="http://web.archive.org/web/20090628095732/http://maths.paisley.ac.uk/allanm/ecrnt/knight/knintro.htm">The Knight's Problem</a>
%H A086446 A. MacLeod, <a href="http://web.archive.org/web/20090226001126/http://maths.paisley.ac.uk/allanm/ECRNT/Ecrnt.htm">Elliptic Curves</a>
%e A086446 a(2)=(1+1+2)*(1/1+1/1+1/2)=10.
%e A086446 a(3)=(1+2+3)*(1/1+1/2+1/3)=6*(11/6)=11.
%e A086446 a(4)=(2+3+10)*(1/2+1/3+1/10)=14.
%e A086446 a(12)=(561+6450+13889)*(1/561+1/6450+1/13889)=42.
%Y A086446 Cf. A085514 (also negative x, y, z admitted).
%K A086446 nonn
%O A086446 1,1
%A A086446 _Hugo Pfoertner_, Jul 19 2003
%E A086446 Corrected and extended by _David J. Rusin_, Jul 30 2003
%E A086446 More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 09 2008