A143574 Sum of all distinct squares occurring when partitioning n into two squares.

0, 1, 1, 0, 4, 5, 0, 0, 4, 9, 10, 0, 0, 13, 0, 0, 16, 17, 9, 0, 20, 0, 0, 0, 0, 50, 26, 0, 0, 29, 0, 0, 16, 0, 34, 0, 36, 37, 0, 0, 40, 41, 0, 0, 0, 45, 0, 0, 0, 49, 75, 0, 52, 53, 0, 0, 0, 0, 58, 0, 0, 61, 0, 0, 64, 130, 0, 0, 68, 0, 0, 0, 36, 73, 74, 0, 0, 0, 0, 0, 80, 81, 82, 0, 0, 170, 0, 0, 0

Offset: 0

Author

Reinhard Zumkeller, Aug 24 2008

Keywords

• nonn

Comments

For n > 0: a(n) = 0 iff A000161(n) = 0: a(A022544(n)) = 0;

A143575 gives numbers m such that a(m) = m.

Examples

			A000161(25)=#{5^2+0^2,4^2+3^2}=2: a(25)=25+0+16+9=50;
A000161(26)=#{5^2+1^2}=1: a(16)=25+1=26;
A000161(49)=#{7^2+0^2}=1: a(49)=49+0=49;
A000161(50)=#{7^2+1^2,5^2+5^2}=2: a(50)=49+1+25=75;
A000161(2600)=#{50^2+10^2,46^2+22^2,38^2+34^2}=3: a(2600)=2500+100+2116+484+1444+1156=7800;
A000161(2601)=#{51^2+0^2,45^2+24^2}=2: a(2601)=2601+0+12025+576=5202;
A000161(2602)=#{51^2+1^2}=1: a(2602)=2601+1=2602.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000161, A002654, A010052, A022544, A143575.

Programs

PARI

a(n) = sum(k=1, n, if (issquare(k) && issquare(n-k), k)); \\ Michel Marcus, May 16 2023

Python

from sympy import divisors
from sympy.solvers.diophantine.diophantine import cornacchia
def A143574(n):
    c = 0
    for d in divisors(n):
        if (k:=d**2)>n:
            break
        q, r = divmod(n,k)
        if not r:
            c += sum(k*(a[0]**2+(a[1]**2 if a[0]!=a[1] else 0)) for a in cornacchia(1,1,q) or [])
    return c # Chai Wah Wu, May 15 2023

Formula

a(n) = Sum_{k=1..n} k*A010052(k)*A010052(n-k). [Reinhard Zumkeller, Sep 27 2008]