OFFSET
1,1
COMMENTS
From Amiram Eldar, Aug 16 2024: (Start)
All the terms are divisible by 6 because sigma_2(k)/k^2 < 3*zeta(2)/4 = 1.2337... < 3/2 for odd numbers k, and sigma_2(k)/k^2 < 8*zeta(2)/9 = 1.462... < 3/2 for numbers k that are not divisible by 3.
There are no 3-smooth numbers (A003586) in this sequence, but for any 5-rough number (A007310) k > 1 there are infinitely many 3-smooth numbers m such that their product k*m is a term.
The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 1, 25, 259, 2578, 25823, 258026, 2580715, 25806329, 258066116, 2580658731, ... . Apparently, the asymptotic density of this sequence exists and equals 0.025806... . (End)
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
FORMULA
A001221(a(n)) >= 3. - Amiram Eldar, Aug 16 2024
EXAMPLE
For k = 60: 1 + 4 + 9 + 16 + 25 + 36 + 100 + 144 + 225 + 400 + 900 + 3600 = 5460 > (3/2) * 3600 = 5400.
MATHEMATICA
Do[s=DivisorSigma[2, n]/(n^2); If[Greater[s, 3/2], Print[n]], {n, 1, 10000}]
Select[Range[2000], DivisorSigma[2, #]/#^2>=3/2&] (* Harvey P. Dale, Mar 05 2013 *)
PROG
(PARI) is(n)=sigma(n, -2) >= 3/2 \\ Charles R Greathouse IV, Feb 03 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Labos Elemer, Jun 25 2004
EXTENSIONS
Name corrected by Charles R Greathouse IV, Feb 03 2018
STATUS
approved