OFFSET
1,1
COMMENTS
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Nov 05 2024
STATUS
approved