A071693 Number of ways to write n as n = x*y*z + x + y + z where 0 <= x <= y <= z <= n.

1, 1, 2, 2, 4, 3, 5, 4, 6, 6, 7, 6, 9, 7, 10, 9, 11, 9, 12, 11, 13, 12, 13, 12, 17, 14, 16, 15, 17, 16, 19, 16, 19, 19, 20, 19, 24, 19, 21, 22, 25, 21, 26, 22, 26, 25, 26, 25, 29, 27, 28, 28, 29, 27, 33, 29, 33, 30, 31, 32, 37, 32, 34, 34, 37, 34, 38, 34, 38, 38, 39, 36, 43, 38, 40

Offset: 0

Author

Benoit Cloitre, Jun 23 2002

Keywords

• easy
• nonn

Comments

a(n) >= floor(n / 2) + 1 = A008619(n). If x = 0 then n = x*y*z+x+y+z = y + z which has floor(n / 2) + 1 solutions. - David A. Corneth, Jul 31 2015

See A260803 for the case where 1 <= x <= y <= z. - M. F. Hasler, Jul 31 2015

Links

• David A. Corneth, Table of n, a(n) for n = 0..9999 (first 1001 terms from Robert G. Wilson v)

Programs

Mathematica

mx = 100; t = 0*Range@ mx; Do[n = x*y*z + x + y + z; If[n < mx, t[[n + 1]]++], {x, 0, mx}, {y, x, mx}, {z, y, mx}]; t (* Robert G. Wilson v, Jul 31 2015 *)

PARI

for(n=0,74,print1(sum(a=0,n,sum(b=0,a,sum(c=0,b,a*b*c+a+b+c==n)))",")) \\ Zak Seidov, Jul 31 2015

PARI

A071693(n)=sum(x=0,n\3,sum(y=x,(n-x*(1+x^2))\2,(n-x-y)%(x*y+1)==0&&n-x>=(x*y+2)*y)) \\ M. F. Hasler, Jul 31 2015

Extensions

a(0) = 1 prepended by David A. Corneth, Jul 30 2015