A391107 Integers k such that the equation x^2 + y^2 = 2*k^2 has at least 4 solutions in positive integers x < y.

65, 85, 130, 145, 170, 185, 195, 205, 221, 255, 260, 265, 290, 305, 325, 340, 365, 370, 377, 390, 410, 425, 435, 442, 445, 455, 481, 485, 493, 505, 510, 520, 530, 533, 545, 555, 565, 580, 585, 595, 610, 615, 625, 629, 650, 663, 680, 685, 689, 697, 715, 725, 730, 740, 745, 754, 765, 780

Offset

1,1

Comments

These numbers are "square-sum-rich" and represent potential centers for a 3 X 3 magic square of distinct squares. Equivalent to n = (x^2 + y^2)/2 with at least 4 distinct pairs (x<y).

If k = Product_i p_i^e_i, where p_i are distinct primes, then k is a term if and only if Product_{i:p_i mod 4 == 1} 2*e_i+1 >= 9. If k is a term, then k*m is a term as well. A131574 is a subsequence. - Chai Wah Wu, Dec 01 2025

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Prog

(Python)
import math
def count_pairs(n):
    L = 2 * n**2
    count = 0
    for x in range(1, int(math.isqrt(L)) + 1):
        y2 = L - x*x
        y = int(math.isqrt(y2))
        if y*y == y2 and y > x:
            count += 1
    return count
# Generate list of n ≤ MAX with ≥4 pairs
MAX = 1500
vals = [n for n in range(1, MAX+1) if count_pairs(n) >= 4]
print(vals)
(Python)
from math import prod
from itertools import count, islice
from sympy import factorint
def A391107_gen(startvalue=1): # generator of terms >= startvalue
    for k in count(max(startvalue, 1)):
        if prod((e<<1)|1 for p, e in factorint(k).items() if p&3==1)>=9:
            yield k
A391107_list = list(islice(A391107_gen(), 58)) # Chai Wah Wu, Dec 01 2025

Crossrefs

Cf. A084648, A002654, A001481, A131574.

Sequence in context: A025303 A071011 A165158 * A084648 A224770 A274044

Adjacent sequences: A391104 A391105 A391106 * A391108 A391109 A391110

Keyword

nonn

Author

Md. Rad Sarar Anando, Nov 29 2025

Status

approved