A377134 Abundant numbers k such that k^2 + A033880(k)^2 is a perfect square.

336, 1080, 3078, 6048, 6552, 19845, 47616, 239760, 435708, 599400, 760320, 873180, 997920, 1468800, 1602300, 2004480, 4312440, 4612608, 4713984, 10181808, 10665984, 11554816, 12160512, 24149664, 31244850, 46431744, 56439504, 64995840, 116958492

Offset: 1

Author

Waldemar Puszkarz, Oct 17 2024

Keywords

• nonn

Comments

These abundant numbers along with their abundances form the legs of an integral Pythagorean triangle.

Odd terms are very rare: 19845 is the only one up to 10^9.

19845 is the only odd term up to 4*10^10. - Amiram Eldar, Mar 11 2025

Examples

			336 is a term because its abundance is 320 and 320^2 + 336^2 = 464^2.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..56

Crossrefs

Cf. A005101, A033880, A000290.

Programs

Mathematica

l={}; Do[a=DivisorSigma[1,n]-2*n; If[a>0&&IntegerQ@Sqrt[n^2+a^2], AppendTo[l, n]], {n, 12, 2*10^8}]; l

PARI

for(n=12, 2*10^8, a=sigma(n)-2*n; a>0&&issquare(n^2+a^2)&&print1(n", "))

Python

import sympy as sp
for i in range(12, 200000000):
    a=sp.ntheory.factor_.divisor_sigma(i) - 2*i
    if a>0 and sp.ntheory.primetest.is_square(i*i+a*a):
        print(i, end=", ")