A395763 Integers x such that there exist two integers 0<x<=y and z>0 such that sigma(x)*psi(x) = sigma(y)*psi(y) = x^2 + y^2 + z^2.

2, 130144, 293760, 1705264, 1977120, 2201280, 6082560, 7012096, 15467520

Offset

1,1

Comments

The numbers x, y and z form a sigma*psi-quadratic triple.

Links

Table of n, a(n) for n=1..9.

S. I. Dimitrov, On σψ-quadratic k-tuples and related generalizations, hal-05303937, 2025.

Example

(293760, 369792, 774144) is such a triple because sigma(293760) * psi(293760) = sigma(369792) * psi(369792) = 1101600 * 746496 = 293760^2 + 369792^2 + 774144^2.

Prog

(Python)
from sympy import divisors, factorint
from functools import lru_cache
from multiprocessing import Pool, cpu_count
from collections import defaultdict
from itertools import combinations_with_replacement
from math import isqrt
from fractions import Fraction
@lru_cache(maxsize=None)
def sigma(n: int) -> int:
    return sum(divisors(n))
@lru_cache(maxsize=None)
def dedekind_psi(n: int) -> Fraction:
    factors = factorint(n)
    result = Fraction(n)
    for p in factors:
        result *= Fraction(p + 1, p)
    return result
def process_n(n):
    return n, sigma(n) * dedekind_psi(n)
def find_xyz_triplets(start, end):
    with Pool(cpu_count()) as pool:
        results = pool.map(process_n, range(start, end + 1))
    groups = defaultdict(list)
    for n, f in results:
        groups[f].append(n)
    triplets = []
    for fval, nums in groups.items():
        for x, y in combinations_with_replacement(nums, 2):
            rhs_frac = fval - x*x - y*y
            if rhs_frac.denominator == 1 and rhs_frac.numerator >= 0:
                z = isqrt(rhs_frac.numerator)
                if z*z == rhs_frac.numerator:
                    triplets.append((x, y, z))
    return triplets
if __name__ == "__main__":
    start = 1
    end = 1000000
    xyz_triplets = find_xyz_triplets(start, end)
    for triplet in xyz_triplets:
        print(triplet)

Crossrefs

Cf. A000203, A000408, A001615, A385356, A395670.

Sequence in context: A170995 A319022 A003840 * A122540 A324438 A167518

Adjacent sequences: A395760 A395761 A395762 * A395764 A395765 A395766

Keyword

nonn,hard,more

Author

S. I. Dimitrov, May 05 2026

Status

approved