A091194 Number of abundant numbers <= n.

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, 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, 12, 12, 13, 13, 13, 13, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 17, 17, 17, 17, 18, 18, 18

Offset: 1

Author

Reinhard Zumkeller, Dec 27 2003

Keywords

• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Marc Deléglise, Bounds for the density of abundant integers, Experiment. Math. Volume 7, Issue 2 (1998), 137-143.
• Charles R. Wall, Phillip L. Crews and Donald B. Johnson, Density bounds for the sum of divisors function, Math. Comp. 26 (1972), 773-777.
• Eric Weisstein's World of Mathematics, Abundant Number

Crossrefs

Partial sums of A294937.

Cf. A005101, A091195, A256562, A302991.

Programs

Magma

[#[k:k in [1..n]| DivisorSigma(1,k) gt 2*k]:n in [1..90]]; // Marius A. Burtea, Nov 06 2019

Mathematica

A091194[1]=0; A091194[n_]:=A091194[n]=A091194[n-1]+Boole[DivisorSigma[1,n]>2n];
(* Recursive expression with data kept in memory; Enrique Pérez Herrero, Aug 14 2010 *)

PARI

a(n) = sum(k=1, n, sigma(k)/k > 2); \\ Michel Marcus, Apr 02 2015

Formula

a(n) ~ c*n, where c = 0.247619... is the asymptotic density of the abundant numbers (A302991). - Amiram Eldar, Mar 21 2021