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A377732 Numbers k such that max{d|k, d <= sqrt(k)} + min{d|k, d >= sqrt(k)} is a square.

%I A377732 #16 Nov 07 2024 08:46:08
%S A377732 3,4,14,18,20,39,46,55,60,63,64,94,114,136,150,154,155,156,158,183,
%T A377732 203,243,258,275,291,295,299,308,315,320,323,324,328,334,444,446,490,
%U A377732 544,558,570,579,580,583,584,588,594,598,600,695,710,718,799,855,878,903,904,938,943,955,959,975,978,979,988,999
%N A377732 Numbers k such that max{d|k, d <= sqrt(k)} + min{d|k, d >= sqrt(k)} is a square.
%C A377732 Numbers k such that A063655(k) = A033676(k) + A033677(k) is a square.
%C A377732 The square terms of this sequence are the positive numbers of the form A141046(m) = 4*m^4.
%H A377732 Amiram Eldar, <a href="/A377732/b377732.txt">Table of n, a(n) for n = 1..10000</a>
%H A377732 Jean-Marie De Koninck, A. Arthur Bonkli Razafindrasoanaivolala, and Hans Schmidt Ramiliarimanana, <a href="https://doi.org/10.1007/s40993-024-00520-x">Integers with a sum of co-divisors yielding a square</a>, Research in Number Theory, Vol. 10, No. 2 (2024), Article 30; <a href="https://www.jeanmariedekoninck.mat.ulaval.ca/fileadmin/Documents/Publications/2023_integers_with_a_sum_of_co-divisors_yielding_a_square.pdf">author's copy</a>.
%F A377732 c * x^(3/4) / log(x) < R(x) < 2 * c * x^(3/4) / log(x) for sufficiently large x, where R(x) is the number of terms that do not exceed x, and c = A377731 (De Koninck et al., 2024).
%t A377732 q[k_] := If[IntegerQ[Sqrt[k]], IntegerQ[Sqrt[2*Sqrt[k]]], Module[{d = Divisors[k], nh}, nh = Length[d]/2; IntegerQ[Sqrt[d[[nh]] + d[[nh + 1]]]]]]; Select[Range[1000], q]
%o A377732 (PARI) is(k) = if(issquare(k), issquare(2 * sqrtint(k)), my(d = divisors(k), nh = #d/2); issquare(d[nh] + d[nh + 1]));
%o A377732 (Python)
%o A377732 from itertools import count, islice
%o A377732 from sympy import divisors
%o A377732 from sympy.ntheory.primetest import is_square
%o A377732 def A377732_gen(startvalue=1): # generator of terms >= startvalue
%o A377732     for k in count(max(startvalue,1)):
%o A377732         d = (a:=divisors(k))[len(a)-1>>1]
%o A377732         if is_square(d+k//d):
%o A377732             yield k
%o A377732 A377732_list = list(islice(A377732_gen(),30)) # _Chai Wah Wu_, Nov 06 2024
%Y A377732 Cf. A033676, A033677, A063655, A377731.
%Y A377732 Subsequences: A141046 \ {0}, A377733, A377736.
%K A377732 nonn
%O A377732 1,1
%A A377732 _Amiram Eldar_, Nov 05 2024