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%I #17 Apr 25 2024 05:18:26
%S 1,2,4,8,12,16,18,24,32,36,48,54,64,72,96,108,128,144,162,192,216,256,
%T 288,324,384,432,486,512,576,648,768,864,972,1024,1152,1296,1458,1536,
%U 1728,1944,2048,2304,2592,2916,3072,3456,3888,4096,4374,4608,5184,5832,6144,6912,7776,8192,8748,9216
%N Numbers k such that sigma(2*k) > 2*k*sqrt(gamma(2*k)), where sigma(k) = A000203(k) is the sum of the divisors of k and gamma(k) = A007947(k) is the greatest squarefree divisor of k.
%C It appears that if sigma(m) > m*sqrt(gamma(m)) then m must be even, and that almost always sigma(m) <= m*sqrt(gamma(m)) (see also A358309).
%C Is there a simpler alternative description of the terms of the present sequence?
%D József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter III, p. 77, section III.1.1.d.
%H Amiram Eldar, <a href="/A358308/b358308.txt">Table of n, a(n) for n = 1..400</a>
%t q[k_] := Module[{f = FactorInteger[2*k], p, e}, {p, e} = Transpose[f]; Times @@ ((p^(e+1)-1)/(p-1)) > 2*k*Sqrt[Times @@ p]]; Select[Range[10^4], q] (* _Amiram Eldar_, Apr 25 2024 *)
%o (PARI) is(k) = {my(f = factor(2*k)); sigma(f)^2 > 4 * k^2 * vecprod(f[, 1]);} \\ _Amiram Eldar_, Apr 25 2024
%Y Cf. A000203, A007947, A358309.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Dec 09 2022
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