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A256562 Number of deficient numbers <= n.

%I A256562 #20 Feb 16 2025 08:33:25
%S A256562 1,2,3,4,5,5,6,7,8,9,10,10,11,12,13,14,15,15,16,16,17,18,19,19,20,21,
%T A256562 22,22,23,23,24,25,26,27,28,28,29,30,31,31,32,32,33,34,35,36,37,37,38,
%U A256562 39,40,41,42,42,43,43,44,45,46,46,47,48,49,50,51,51,52,53
%N A256562 Number of deficient numbers <= n.
%H A256562 Amiram Eldar, <a href="/A256562/b256562.txt">Table of n, a(n) for n = 1..10000</a>
%H A256562 Marc Deléglise, <a href="http://projecteuclid.org/euclid.em/1048515661">Bounds for the density of abundant integers</a>, Experiment. Math. Volume 7, Issue 2 (1998), 137-143.
%H A256562 Charles R. Wall, Phillip L. Crews and Donald B. Johnson, <a href="http://dx.doi.org/10.1090/S0025-5718-1972-0327700-7">Density bounds for the sum of divisors function</a>, Math. Comp. 26 (1972), 773-777.
%H A256562 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AbundantNumber.html">Abundant Number</a>
%F A256562 a(n) ~ c*n, where c = 0.752380... is the asymptotic density of the deficient numbers (A318172). - _Amiram Eldar_, Mar 21 2021
%e A256562 For k=1 to 5, all numbers are deficients so a(k) = k in this range.
%e A256562 a(6) = 5 since 6 is the first number that is not deficient.
%t A256562 a[n_]:=Length[Select[Range[n],DivisorSigma[1,#]/#<2&]];a/@Range[68] (* _Ivan N. Ianakiev_, Apr 03 2015 *)
%o A256562 (PARI) a(n) = sum(k=1, n, sigma(k)/k < 2);
%o A256562 (Magma) [#[k:k in [1..n]| DivisorSigma(1,k) lt 2*k]:n in [1..70]]; // _Marius A. Burtea_, Nov 06 2019
%Y A256562 Partial sums of A294934.
%Y A256562 Cf. A000396 (perfect), A005100 (deficient), A005101 (abundant).
%Y A256562 Cf. A091194 (number of abundant numbers <= n).
%Y A256562 Cf. A256440, A318172.
%K A256562 nonn
%O A256562 1,2
%A A256562 _Michel Marcus_, Apr 02 2015