A256562 Number of deficient numbers <= n.

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, 22, 22, 23, 23, 24, 25, 26, 27, 28, 28, 29, 30, 31, 31, 32, 32, 33, 34, 35, 36, 37, 37, 38, 39, 40, 41, 42, 42, 43, 43, 44, 45, 46, 46, 47, 48, 49, 50, 51, 51, 52, 53

Offset

1,2

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

Formula

a(n) ~ c*n, where c = 0.752380... is the asymptotic density of the deficient numbers (A318172). - Amiram Eldar, Mar 21 2021

Example

For k=1 to 5, all numbers are deficients so a(k) = k in this range.
a(6) = 5 since 6 is the first number that is not deficient.

Mathematica

a[n_]:=Length[Select[Range[n], DivisorSigma[1, #]/#<2&]]; a/@Range[68] (* Ivan N. Ianakiev, Apr 03 2015 *)

Prog

(PARI) a(n) = sum(k=1, n, sigma(k)/k < 2);
(Magma) [#[k:k in [1..n]| DivisorSigma(1, k) lt 2*k]:n in [1..70]]; // Marius A. Burtea, Nov 06 2019

Crossrefs

Partial sums of A294934.

Cf. A000396 (perfect), A005100 (deficient), A005101 (abundant).

Cf. A091194 (number of abundant numbers <= n).

Cf. A256440, A318172.

Sequence in context: A006163 A331268 A053757 * A228297 A303788 A366871

Adjacent sequences: A256559 A256560 A256561 * A256563 A256564 A256565

Keyword

nonn

Author

Michel Marcus, Apr 02 2015

Status

approved