A082986 Largest x such that 1/x + 1/y + 1/z = 1/n.

6, 42, 156, 420, 930, 1806, 3192, 5256, 8190, 12210, 17556, 24492, 33306, 44310, 57840, 74256, 93942, 117306, 144780, 176820, 213906, 256542, 305256, 360600, 423150, 493506, 572292, 660156, 757770, 865830, 985056, 1116192, 1260006, 1417290, 1588860, 1775556

Offset

1,1

Comments

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

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..5000

Formula

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]

a(n) >= 6*A006325(n-1). - Robert G. Wilson v, May 04 2013 [Corrected by Michael Somos, Jan 27 2014]

a(n) < 3*n^4 for n>=2 (upper bound). - Carmine Suriano, Feb 20 2014

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A082604, A006325, A169938.

Sequence in context: A360757 A176780 A169938 * A180806 A253946 A359847

Adjacent sequences: A082983 A082984 A082985 * A082987 A082988 A082989

Keyword

nonn,nice

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), May 29 2003

Extensions

a(6)-a(36) from Charles R Greathouse IV, Oct 17 2012

Deleted incorrect (or at least unproved) Mma program. - N. J. A. Sloane, Jan 27 2014

Status

approved