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

%I A377134 #16 Mar 11 2025 04:39:45
%S A377134 336,1080,3078,6048,6552,19845,47616,239760,435708,599400,760320,
%T A377134 873180,997920,1468800,1602300,2004480,4312440,4612608,4713984,
%U A377134 10181808,10665984,11554816,12160512,24149664,31244850,46431744,56439504,64995840,116958492
%N A377134 Abundant numbers k such that k^2 + A033880(k)^2 is a perfect square.
%C A377134 These abundant numbers along with their abundances form the legs of an integral Pythagorean triangle.
%C A377134 Odd terms are very rare: 19845 is the only one up to 10^9.
%C A377134 19845 is the only odd term up to 4*10^10. - _Amiram Eldar_, Mar 11 2025
%H A377134 Amiram Eldar, <a href="/A377134/b377134.txt">Table of n, a(n) for n = 1..56</a>
%e A377134 336 is a term because its abundance is 320 and 320^2 + 336^2 = 464^2.
%t A377134 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
%o A377134 (PARI) for(n=12, 2*10^8, a=sigma(n)-2*n; a>0&&issquare(n^2+a^2)&&print1(n", "))
%o A377134 (Python)
%o A377134 import sympy as sp
%o A377134 for i in range(12, 200000000):
%o A377134     a=sp.ntheory.factor_.divisor_sigma(i) - 2*i
%o A377134     if a>0 and sp.ntheory.primetest.is_square(i*i+a*a):
%o A377134         print(i, end=", ")
%Y A377134 Cf. A005101, A033880, A000290.
%K A377134 nonn
%O A377134 1,1
%A A377134 _Waldemar Puszkarz_, Oct 17 2024