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%I #41 Jul 10 2017 05:53:00
%S 6,42,156,420,930,1806,3192,5256,8190,12210,17556,24492,33306,44310,
%T 57840,74256,93942,117306,144780,176820,213906,256542,305256,360600,
%U 423150,493506,572292,660156,757770,865830,985056,1116192,1260006,1417290,1588860,1775556
%N Largest x such that 1/x + 1/y + 1/z = 1/n.
%C The greedy algorithm gives the decomposition 1/n = 1/(n+1) + 1/(n^2+n+1) + 1/(n^4+2n^3+2n^2+n). - _Charles R Greathouse IV_, Oct 17 2012
%H Charles R Greathouse IV, <a href="/A082986/b082986.txt">Table of n, a(n) for n = 1..5000</a>
%F a(n) >= n^4 + 2n^3 + 2n^2 + n (cf. A169938). - _Charles R Greathouse IV_, Oct 17 2012. [Note this is at present only a lower bound! - _N. J. A. Sloane_, Jan 27 2014]
%F a(n) >= 6*A006325(n-1). - _Robert G. Wilson v_, May 04 2013 [Corrected by _Michael Somos_, Jan 27 2014]
%F a(n) < 3*n^4 for n>=2 (upper bound). - _Carmine Suriano_, Feb 20 2014
%t a[n_] := Module[{f, d, t, x = 0}, For[z = n+1, z <= Quotient[201*n, 100], z++, f = 1/n - 1/z; d = Denominator[f]; Do[t = (y/d + 1/y)/f; If[Denominator[t] == 1, x = Max[x, t*y]], {y, Divisors[d]}]]; x]; Table[a[n], {n, 1, 36}] (* _Jean-François Alcover_, Jul 10 2017, after _Charles R Greathouse IV_ *)
%o (PARI) a(n)=my(f, d, t, x); for(z=n+1, 201*n\100, f=1/n-1/z; d=denominator(f); fordiv(d, y, t=(y/d+1/y)/f; if(denominator(t)==1, x=max(x, t*y)))); x \\ _Charles R Greathouse IV_, Oct 17 2012
%Y Cf. A082604, A006325, A169938.
%K nonn,nice
%O 1,1
%A Yuval Dekel (dekelyuval(AT)hotmail.com), May 29 2003
%E a(6)-a(36) from _Charles R Greathouse IV_, Oct 17 2012
%E Deleted incorrect (or at least unproved) Mma program. - _N. J. A. Sloane_, Jan 27 2014
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