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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.
2
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
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
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Dec 09 2022
STATUS
approved