A385356 Numbers x such that there exist two integers 0<x<=y and z>0 such that sigma(x)^2 = sigma(y)^2 = x^2 + y^2 + z^2.

2, 40, 164, 196, 224, 1120, 3040, 13440, 22932, 44200, 76160, 90848, 91720, 174592, 530200, 619840, 687184, 872960, 1686400, 1767040, 1807120, 1927680, 1990912, 2154880, 3653760, 4286880, 5637632, 5759680, 6442128, 8225280, 8943800, 9264320, 9465600, 9694080

Offset

1,1

Comments

The numbers x, y and z form a sigma-quadratic triple. See Dimitrov link.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..161

S. I. Dimitrov, Generalizations of amicable numbers, arXiv:2408.07387 [math.NT], 2024.

Example

(40, 58, 56) is such a triple because sigma(40)^2 = sigma(58)^2 = 90^2 = 40^2 + 58^2 + 56^2.

Prog

(Python)
from itertools import count, islice
from sympy import divisor_sigma
from sympy.ntheory.primetest import is_square
def A385356_gen(startvalue=1): # generator of terms >= startvalue
    for x in count(max(startvalue, 1)):
        sx, x2 = int(divisor_sigma(x)), x**2
        sx2 = sx**2
        if sx2>x2:
            for y in count(x):
                if (k:=sx2-x2-y**2)<=0:
                    break
                if is_square(k) and sx==divisor_sigma(y):
                    yield x
                    break
A385356_list = list(islice(A385356_gen(), 8)) # Chai Wah Wu, Jul 02 2025

Crossrefs

Cf. A000203, A066784, A096907, A096908, A096909, A096910, A385325.

Sequence in context: A389884 A281761 A370069 * A359859 A012477 A003760

Adjacent sequences: A385353 A385354 A385355 * A385357 A385358 A385359

Keyword

nonn

Author

S. I. Dimitrov, Jun 26 2025

Extensions

Data corrected by David A. Corneth, Jun 27 2025

a(18)-a(34) from Chai Wah Wu, Jul 02 2025

Status

approved