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a(n) = Sum_{b=0..floor(sqrt(n)), n-b^2 is square} b.
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%I #35 May 16 2023 12:53:27

%S 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,

%T 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,

%U 0,10,0,0,0,6,11,12,0,0,0,0,0,12,9,10,0

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

%C a(n) = 0 if n in A022544.

%C a(n) > 0 if n in A001481.

%H Stefano Spezia, <a href="/A362961/b362961.txt">Table of n, a(n) for n = 1..10000</a>

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

%o (Python)

%o from gmpy2 import *

%o a = lambda n: sum([b for b in range(0, isqrt(n) + 1) if is_square(n - (b*b))])

%o print([a(n) for n in range(1, 84)])

%o (Python)

%o from sympy import divisors

%o from sympy.solvers.diophantine.diophantine import cornacchia

%o def A362961(n):

%o c = 0

%o for d in divisors(n):

%o if (k:=d**2)>n:

%o break

%o q, r = divmod(n,k)

%o if not r:

%o c += sum(d*(a[0]+(a[1] if a[0]!=a[1] else 0)) for a in cornacchia(1,1,q) or [])

%o return c # _Chai Wah Wu_, May 15 2023

%o (PARI) a(n) = sum(b=0, sqrtint(n), if (issquare(n-b^2), b)); \\ _Michel Marcus_, May 16 2023

%Y Cf. A022544, A001481.

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

%K nonn,look

%O 1,4

%A _DarĂ­o Clavijo_, May 10 2023