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

0,5

Comments

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

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

Links

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

Formula

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

Example

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.

Prog

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

Crossrefs

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

Sequence in context: A167003 A335762 A254294 * A262400 A075424 A200619

Adjacent sequences: A143571 A143572 A143573 * A143575 A143576 A143577

Keyword

nonn

Author

Reinhard Zumkeller, Aug 24 2008

Status

approved