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A096490
Numbers k such that sigma_2(k) >= (3/2) * k^2, where sigma_2(k) is the sum of the squares of the divisors of k.
2
60, 120, 168, 180, 240, 252, 300, 336, 360, 420, 480, 504, 540, 600, 660, 672, 720, 756, 780, 792, 840, 900, 924, 936, 960, 1008, 1020, 1080, 1140, 1176, 1200, 1260, 1320, 1344, 1380, 1440, 1500, 1512, 1560, 1584, 1620, 1680, 1740, 1764, 1800, 1848, 1860
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
Cf. A001157, A056866, A118671 (primitive terms).
Sequence in context: A252962 A296767 A336442 * A056866 A098136 A060793
KEYWORD
nonn,easy
AUTHOR
Labos Elemer, Jun 25 2004
EXTENSIONS
Name corrected by Charles R Greathouse IV, Feb 03 2018
STATUS
approved