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
Keywords
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.
Programs
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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] -
PARI
is(k) = if(issquare(k), issquare(2 * sqrtint(k)), my(d = divisors(k), nh = #d/2); issquare(d[nh] + d[nh + 1]));
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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
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).
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