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

%I A067808 #32 Apr 27 2025 09:08:27
%S A067808 720,1080,1440,1680,1800,2016,2160,2520,2880,3024,3240,3360,3600,3780,
%T A067808 3960,4032,4200,4320,4680,5040,5280,5400,5544,5760,6048,6120,6300,
%U A067808 6480,6720,6840,7056,7200,7560,7920,8064,8400,8640,9000,9072,9240,9360,9504
%N A067808 Numbers k such that sigma(k)^2 > 3*sigma(k^2).
%C A067808 For every m>1 sigma(m)^2 > sigma(m^2).
%C A067808 From _Robert Israel_, Jun 20 2018: (Start)
%C A067808 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.
%C A067808 If h is a term then so are all multiples of h.
%C A067808 The first term that is squarefree is 7420738134810 = A002110(12). (End)
%C A067808 From _Amiram Eldar_, Apr 27 2025: (Start)
%C A067808 All the terms are 3-abundant numbers (A068403).
%C A067808 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)
%H A067808 Robert Israel, <a href="/A067808/b067808.txt">Table of n, a(n) for n = 1..10000</a>
%p A067808 filter:= proc(n) local F;
%p A067808   F:= ifactors(n)[2];
%p A067808   mul((t[1]^(t[2]+1)-1)^2/(t[1]^(2*t[2]+1)-1)/(t[1]-1), t = F) > 3
%p A067808 end proc:
%p A067808 select(filter, [$1..10^4]); # _Robert Israel_, Jun 20 2018
%t A067808 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];
%t A067808 Select[Range[10^4], filterQ] (* _Jean-François Alcover_, Apr 29 2019, after _Robert Israel_ *)
%o A067808 (PARI) isok(k) = sigma(k)^2 > 3*sigma(k^2); \\ _Michel Marcus_, Apr 29 2019
%o A067808 (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
%Y A067808 Cf. A002110, A065764.
%Y A067808 Subsequence of A068403.
%K A067808 nonn
%O A067808 1,1
%A A067808 _Benoit Cloitre_, Feb 07 2002