A084888 Number of partitions of n^3 into two squares>0.

0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 0, 2, 1, 0, 2, 0, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 0, 4, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 8, 0, 0, 2, 0, 0, 0, 1, 2, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 8, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 1, 0, 3, 2

Offset: 0

Author

Reinhard Zumkeller, Jun 10 2003

Keywords

• nonn

Comments

a(A050804(n)) = 1.

Examples

			n=100: 100^3 = 1000000 = 960^2 + 280^2 = 936^2 + 352^2 = 800^2 + 600^2, therefore a(100)=3.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Diophantine Equation: 2nd Powers.
• Reinhard Zumkeller, Illustration for A084888 and A000404

Crossrefs

Cf. A000404, A063725, A000578, A000290.

Programs

Haskell

a084888 = a025426 . a000578  -- Reinhard Zumkeller, Jul 18 2012

PARI

a(n)=my(f=factor(n^3)); (prod(i=1,#f~,if(f[i,1]%4==1,f[i,2]+1,f[i,2]%2==0||f[i,1]<3))-issquare(n)+1)\2 \\ Charles R Greathouse IV, May 18 2016

Python

from math import prod
from sympy import factorint
def A084888(n): return ((m:=prod(1 if p==2 else (3*e+1 if p&3==1 else (3*e+1)&1) for p, e in factorint(n).items()))+((((~n**3 & n**3-1).bit_length()&1)<<1)-1 if m&1 else 0))>>1 # Chai Wah Wu, May 17 2023

Formula

a(n) = A025426(A000578(n)).