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.

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

Author

Labos Elemer, Jun 25 2004

Keywords

• nonn
• easy

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)

Examples

			For k = 60: 1 + 4 + 9 + 16 + 25 + 36 + 100 + 144 + 225 + 400 + 900 + 3600 = 5460 > (3/2) * 3600 = 5400.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001157, A056866, A118671 (primitive terms).

Cf. A001221, A003586, A007310, A013661.

Programs

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 *)

PARI

is(n)=sigma(n,-2) >= 3/2 \\ Charles R Greathouse IV, Feb 03 2018

Formula

A001221(a(n)) >= 3. - Amiram Eldar, Aug 16 2024

Extensions

Name corrected by Charles R Greathouse IV, Feb 03 2018