A362961 a(n) = Sum_{b=0..floor(sqrt(n)), n-b^2 is square} b.

1, 1, 0, 2, 3, 0, 0, 2, 3, 4, 0, 0, 5, 0, 0, 4, 5, 3, 0, 6, 0, 0, 0, 0, 12, 6, 0, 0, 7, 0, 0, 4, 0, 8, 0, 6, 7, 0, 0, 8, 9, 0, 0, 0, 9, 0, 0, 0, 7, 13, 0, 10, 9, 0, 0, 0, 0, 10, 0, 0, 11, 0, 0, 8, 20, 0, 0, 10, 0, 0, 0, 6, 11, 12, 0, 0, 0, 0, 0, 12, 9, 10, 0

Offset: 1

Author

Darío Clavijo, May 10 2023

Keywords

• nonn
• look

Comments

a(n) = 0 if n in A022544.

a(n) > 0 if n in A001481.

Links

• Stefano Spezia, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A022544, A001481.

Cf. A143574 (sum of b^2), A000925.

Programs

Mathematica

a[n_]:=Sum[b Boole[IntegerQ[Sqrt[n-b^2]]],{b,0,Floor[Sqrt[n]]}]; Array[a,83] (* Stefano Spezia, May 15 2023 *)

PARI

a(n) = sum(b=0, sqrtint(n), if (issquare(n-b^2), b)); \\ Michel Marcus, May 16 2023

Python

from gmpy2 import *
a = lambda n: sum([b for b in range(0, isqrt(n) + 1) if is_square(n - (b*b))])
print([a(n) for n in range(1, 84)])

Python

from sympy import divisors
from sympy.solvers.diophantine.diophantine import cornacchia
def A362961(n):
    c = 0
    for d in divisors(n):
        if (k:=d**2)>n:
            break
        q, r = divmod(n,k)
        if not r:
            c += sum(d*(a[0]+(a[1] if a[0]!=a[1] else 0)) for a in cornacchia(1,1,q) or [])
    return c # Chai Wah Wu, May 15 2023