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

3, 4, 14, 18, 20, 39, 46, 55, 60, 63, 64, 94, 114, 136, 150, 154, 155, 156, 158, 183, 203, 243, 258, 275, 291, 295, 299, 308, 315, 320, 323, 324, 328, 334, 444, 446, 490, 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

Offset

1,1

Comments

Numbers k such that A063655(k) = A033676(k) + A033677(k) is a square.

The square terms of this sequence are the positive numbers of the form A141046(m) = 4*m^4.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Jean-Marie De Koninck, A. Arthur Bonkli Razafindrasoanaivolala, and Hans Schmidt Ramiliarimanana, Integers with a sum of co-divisors yielding a square, Research in Number Theory, Vol. 10, No. 2 (2024), Article 30; author's copy.

Formula

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).

Mathematica

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]

Prog

(PARI) is(k) = if(issquare(k), issquare(2 * sqrtint(k)), my(d = divisors(k), nh = #d/2); issquare(d[nh] + d[nh + 1]));
(Python)
from itertools import count, islice
from sympy import divisors
from sympy.ntheory.primetest import is_square
def A377732_gen(startvalue=1): # generator of terms >= startvalue
    for k in count(max(startvalue, 1)):
        d = (a:=divisors(k))[len(a)-1>>1]
        if is_square(d+k//d):
            yield k
A377732_list = list(islice(A377732_gen(), 30)) # Chai Wah Wu, Nov 06 2024

Crossrefs

Cf. A033676, A033677, A063655, A377731.

Subsequences: A141046 \ {0}, A377733, A377736.

Sequence in context: A282307 A282604 A281524 * A299040 A297839 A279938

Adjacent sequences: A377729 A377730 A377731 * A377733 A377734 A377735

Keyword

nonn

Author

Amiram Eldar, Nov 05 2024

Status

approved