A290870 - OEIS

%I #30 Feb 16 2024 01:30:41

%S 0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,1,1,0,0,2,0,0,2,1,0,2,0,2,1,0,1,

%T 2,1,0,2,2,0,3,0,1,3,0,1,4,0,1,2,2,1,2,2,2,3,0,0,5,0,2,3,2,1,2,2,1,4,

%U 2,0,6,0,1,4,2,3,2,0,4,3,2,1,5,2,0,4,4,0,5,2,2,4,0,3,6,2,1,3,3,1

%N a(n) is the number of ways to represent n as n = x*y + y*z + z*x where 0 < x < y < z.

%C a(n) = 0 if and only if n is a term of A000926.

%C a(n) = 1 if and only if n is a term of A093669.

%H Joerg Arndt, <a href="/A290870/b290870.txt">Table of n, a(n) for n = 1..10000</a>

%F For the triples (x,y,z) we have x < sqrt(n / 3), y < (n - x^2) / (2 * x), z = (n - x*y) / (x + y) which must be integer. - _David A. Corneth_, Oct 01 2017

%e For (x, y, z) = (1, 3, 5), we have x * y + y * z + z * x = 1 * 3 + 3 * 5 + 5 * 1 = 23 and similarily for (x, y, z) = (1, 2, 7), we have x * y + y * z + z * x = 23. Those 2 triples are all for n=23, so a(23) = 2. - _David A. Corneth_, Oct 01 2017

%o (PARI) N=10^3; V=vector(N);

%o { for (x=1, N,

%o     for (y=x+1, N, t=x*y; if( t > N, break() );

%o       for (z=y+1, N,

%o         tt = t + y*z + z*x;  if( tt > N, break() );

%o         V[tt]+=1;

%o ); ); ); }

%o V \\ _Joerg Arndt_, Oct 01 2017

%o (PARI) a(n) = {my(res = 0);

%o for(x = 1, sqrtint(n\3), for(y = x + 1, (n - x^2) \ (2 * x), z = (n - x*y) / (x + y); if(z > y && z == z\1, res++))); res} \\ _David A. Corneth_, Oct 01 2017

%Y Cf. A066955 (ways to represent n as n = x*y + y*z + z*x where 0 <= x <= y <= z).

%Y Cf. A094377 (greatest number having exactly n representations).

%Y Cf. A094376 (indices of records).

%K nonn

%O 1,23

%A _Joerg Arndt_, Aug 13 2017