A290870 a(n) is the number of ways to represent n as n = x*y + y*z + z*x where 0 < x < y < z.

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, 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, 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

Offset

1,23

Comments

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

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

Links

Joerg Arndt, Table of n, a(n) for n = 1..10000

Formula

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

Example

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

Prog

(PARI) N=10^3; V=vector(N);
{ for (x=1, N,
    for (y=x+1, N, t=x*y; if( t > N, break() );
      for (z=y+1, N,
        tt = t + y*z + z*x;  if( tt > N, break() );
        V[tt]+=1;
); ); ); }
V \\ Joerg Arndt, Oct 01 2017
(PARI) a(n) = {my(res = 0);
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

Crossrefs

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

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

Cf. A094376 (indices of records).

Sequence in context: A238304 A219487 A303907 * A353501 A353428 A244738

Adjacent sequences: A290867 A290868 A290869 * A290871 A290872 A290873

Keyword

nonn

Author

Joerg Arndt, Aug 13 2017

Status

approved