A358308 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.

1, 2, 4, 8, 12, 16, 18, 24, 32, 36, 48, 54, 64, 72, 96, 108, 128, 144, 162, 192, 216, 256, 288, 324, 384, 432, 486, 512, 576, 648, 768, 864, 972, 1024, 1152, 1296, 1458, 1536, 1728, 1944, 2048, 2304, 2592, 2916, 3072, 3456, 3888, 4096, 4374, 4608, 5184, 5832, 6144, 6912, 7776, 8192, 8748, 9216

Offset

1,2

Comments

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).

Is there a simpler alternative description of the terms of the present sequence?

References

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.

Links

Amiram Eldar, Table of n, a(n) for n = 1..400

Mathematica

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 *)

Prog

(PARI) is(k) = {my(f = factor(2*k)); sigma(f)^2 > 4 * k^2 * vecprod(f[, 1]); } \\ Amiram Eldar, Apr 25 2024

Crossrefs

Cf. A000203, A007947, A358309.

Sequence in context: A124395 A334167 A024908 * A325326 A215459 A019442

Adjacent sequences: A358305 A358306 A358307 * A358309 A358310 A358311

Keyword

nonn

Author

N. J. A. Sloane, Dec 09 2022

Status

approved