OFFSET
0,6
COMMENTS
a(A018825(n))=0; a(A000404(n))>0; a(A081324(n))=1; a(A004431(n))>1. - Reinhard Zumkeller, Aug 16 2011
LINKS
FORMULA
G.f.: (Sum_{m=1..inf} x^(m^2))^2.
G.f.: (theta_3(q) - 1)^2/4, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 08 2018
From Ridouane Oudra, Dec 08 2025: (Start) for n > 0 :
a(n) = Sum_{k=1..floor(sqrt(n-1))} A010052(n-k^2).
a(2*n-1) = 2*A025441(2*n-1).
a(2*n) = a(n) + A143259(n).
a(2^k*n) = a(n) + (k mod 2)*A143259(n), for all k >= 0 and n > 0. (End)
EXAMPLE
a(5) = 2 from the solutions (1,2) and (2,1).
MATHEMATICA
nn = 100; t = Table[0, {nn}]; s = Sqrt[nn]; Do[n = x^2 + y^2; If[n <= nn, t[[n]]++], {x, s}, {y, s}]; Join[{0}, t] (* T. D. Noe, Apr 03 2011 *)
PROG
(Haskell)
a063725 n =
sum $ map (a010052 . (n -)) $ takeWhile (< n) $ tail a000290_list
a063725_list = map a063725 [0..]
-- Reinhard Zumkeller, Aug 16 2011
(PARI) a(n)=if(n==0, return(0)); my(f=factor(n)); prod(i=1, #f~, if(f[i, 1]%4==1, f[i, 2]+1, f[i, 2]%2==0 || f[i, 1]==2)) - issquare(n) \\ Charles R Greathouse IV, May 18 2016
(Python)
from math import prod
from sympy import factorint
def A063725(n):
f = factorint(n)
return prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in f.items())-(not any(e&1 for e in f.values())) if n else 0 # Chai Wah Wu, May 17 2023
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Aug 23 2001
STATUS
approved