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A091194 Number of abundant numbers <= n.

%I A091194 #24 Feb 16 2025 08:32:52
%S A091194 0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,
%T A091194 5,6,6,6,6,7,7,8,8,8,8,8,8,9,9,9,9,9,9,10,10,11,11,11,11,12,12,12,12,
%U A091194 12,12,13,13,13,13,14,14,15,15,15,15,15,15,16,16,17,17,17,17,18,18,18
%N A091194 Number of abundant numbers <= n.
%H A091194 Amiram Eldar, <a href="/A091194/b091194.txt">Table of n, a(n) for n = 1..10000</a>
%H A091194 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 A091194 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 A091194 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AbundantNumber.html">Abundant Number</a>
%F A091194 a(n) ~ c*n, where c = 0.247619... is the asymptotic density of the abundant numbers (A302991). - _Amiram Eldar_, Mar 21 2021
%t A091194 A091194[1]=0; A091194[n_]:=A091194[n]=A091194[n-1]+Boole[DivisorSigma[1,n]>2n];
%t A091194 (* Recursive expression with data kept in memory; _Enrique Pérez Herrero_, Aug 14 2010 *)
%o A091194 (PARI) a(n) = sum(k=1, n, sigma(k)/k > 2); \\ _Michel Marcus_, Apr 02 2015
%o A091194 (Magma) [#[k:k in [1..n]| DivisorSigma(1,k) gt 2*k]:n in [1..90]]; // _Marius A. Burtea_, Nov 06 2019
%Y A091194 Partial sums of A294937.
%Y A091194 Cf. A005101, A091195, A256562, A302991.
%K A091194 nonn
%O A091194 1,18
%A A091194 _Reinhard Zumkeller_, Dec 27 2003