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A049692 Numbers k such that the square of the number of divisors of k does not divide the sum of the divisors of k.

%I A049692 #20 Apr 25 2024 09:12:10
%S A049692 2,4,5,6,8,9,10,12,13,14,15,16,17,18,20,22,24,25,26,27,28,29,30,32,34,
%T A049692 36,37,38,39,40,41,42,44,45,46,48,49,50,51,52,53,54,55,56,58,60,61,63,
%U A049692 64,65,66,68,70,72,73,74,75,76,78,80,81,82,84,85,86
%N A049692 Numbers k such that the square of the number of divisors of k does not divide the sum of the divisors of k.
%C A049692 The asymptotic density of this sequence is 1/2 (Bateman et al., 1981). - _Amiram Eldar_, Apr 25 2024
%D A049692 József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter III, p. 119, section III.51.
%H A049692 Amiram Eldar, <a href="/A049692/b049692.txt">Table of n, a(n) for n = 1..10000</a>
%H A049692 Paul T. Bateman, Paul Erdős, Carl Pomerance and E. G. Straus, <a href="https://doi.org/10.1007/BFb0096462">The arithmetic mean of the divisors of an integer</a>, in Marvin I. Knopp (ed.), Analytic Number Theory, Proceedings of a Conference Held at Temple University, Philadelphia, May 12-15, 1980, Lecture Notes in Mathematics, Vol. 899, Springer, Berlin - New York, 1981, pp. 197-220, <a href="https://math.dartmouth.edu/~carlp/PDF/31.pdf">alternative link</a>.
%p A049692 with(numtheory): t := [ ]: f := [ ]: for n from 1 to 500 do if sigma(n) mod sigma[ 0 ](n)^2 = 0 then t := [ op(t), n ] else f := [ op(f), n ]; fi; od: f;
%t A049692 Select[Range[1, 100], !Divisible[DivisorSigma[1, #], DivisorSigma[0, #]^2] &] (* _Amiram Eldar_, Apr 25 2024 *)
%o A049692 (PARI) is(n) = {my(f = factor(n)); sigma(f) % numdiv(f)^2 > 0;} \\ _Amiram Eldar_, Apr 25 2024
%Y A049692 Cf. A000005, A000203, A003601, A049642, A049645 (complement).
%K A049692 nonn
%O A049692 1,1
%A A049692 _N. J. A. Sloane_