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Number of ways to write n as n = x*y*z/(x+y+z) 1 <= x <= y <= z <= n.
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%I #22 Oct 12 2020 03:45:28

%S 0,0,1,0,1,0,1,2,2,1,2,3,1,3,5,3,2,3,2,7,5,1,3,8,2,4,8,4,3,5,2,9,5,2,

%T 14,12,1,4,10,10,3,5,2,11,11,2,4,13,5,10,8,6,3,14,12,22,5,1,5,14,1,5,

%U 22,11,13,11,2,11,9,6,5,19,1,5,19,10,13,6,4,33,14,1,5,24,6,6,11,10,5

%N Number of ways to write n as n = x*y*z/(x+y+z) 1 <= x <= y <= z <= n.

%C If n is a prime other than 3, then a(n) = A000005(n+1)/2 - 1. - _Robert Israel_, Oct 29 2018

%H Robert Israel, <a href="/A071694/b071694.txt">Table of n, a(n) for n = 1..1000</a>

%p f:= proc(n) local x,y,z,t;

%p t:= 0;

%p for x from 1 to n do

%p for y from max(x,ceil(n/x)) to n do

%p if x*y = n then next fi;

%p z:= n*(x+y)/(x*y-n);

%p if z::integer and z>=y and z<=n then t:= t+1 fi

%p od od:

%p t

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, Oct 29 2018

%t a[n_] := Sum[Boole[x y z/(x + y + z) == n], {x, n}, {y, x}, {z, y}];

%t Array[a, 100] (* _Jean-François Alcover_, Aug 24 2020 *)

%o (PARI) for(n=1,90,print1(sum(a=1,n,sum(b=1,a,sum(c=1,b,if(a*b*c/(a+b+c)-n,0,1)))),","))

%Y Cf. A000005, A063520.

%K nonn

%O 1,8

%A _Benoit Cloitre_, Jun 23 2002