A067808 Numbers k such that sigma(k)^2 > 3*sigma(k^2).

720, 1080, 1440, 1680, 1800, 2016, 2160, 2520, 2880, 3024, 3240, 3360, 3600, 3780, 3960, 4032, 4200, 4320, 4680, 5040, 5280, 5400, 5544, 5760, 6048, 6120, 6300, 6480, 6720, 6840, 7056, 7200, 7560, 7920, 8064, 8400, 8640, 9000, 9072, 9240, 9360, 9504

Offset: 1

Author

Benoit Cloitre, Feb 07 2002

Keywords

• nonn

Comments

For every m>1 sigma(m)^2 > sigma(m^2).

From Robert Israel, Jun 20 2018: (Start)

Numbers with prime factorization Product_j p_j^(e_j) such that Product_j (p_j^(e_j+1)-1)^2/((p_j^(2*e_j+1)-1)*(p_j-1)) > 3.

If h is a term then so are all multiples of h.

The first term that is squarefree is 7420738134810 = A002110(12). (End)

From Amiram Eldar, Apr 27 2025: (Start)

All the terms are 3-abundant numbers (A068403).

The numbers of terms that do not exceed 10^k, for k = 3, 4, ..., are 1, 44, 501, 5246, 51870, 518782, 5191909, 51889993, 518783441, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00518... . (End)

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002110, A065764.

Subsequence of A068403.

Programs

Maple

filter:= proc(n) local F;
  F:= ifactors(n)[2];
  mul((t[1]^(t[2]+1)-1)^2/(t[1]^(2*t[2]+1)-1)/(t[1]-1), t = F) > 3
end proc:
select(filter, [$1..10^4]); # Robert Israel, Jun 20 2018

Mathematica

filterQ[n_] := Module[{F = FactorInteger[n]}, If[n == 1, Return[False]]; Product[{p, e} = pe; (p^(e+1)-1)^2/((p^(2e+1)-1)(p-1)), {pe, F}] > 3];
Select[Range[10^4], filterQ] (* Jean-François Alcover, Apr 29 2019, after Robert Israel *)

PARI

isok(k) = sigma(k)^2 > 3*sigma(k^2); \\ Michel Marcus, Apr 29 2019

PARI

isok(k) = {my(f = factor(k), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; (p^(e+1)-1)^2/((p^(2*e+1)-1)*(p-1))) > 3;} \\ Amiram Eldar, Apr 27 2025