

A033880


Abundance of n, or (sum of divisors of n)  2n.


109



1, 1, 2, 1, 4, 0, 6, 1, 5, 2, 10, 4, 12, 4, 6, 1, 16, 3, 18, 2, 10, 8, 22, 12, 19, 10, 14, 0, 28, 12, 30, 1, 18, 14, 22, 19, 36, 16, 22, 10, 40, 12, 42, 4, 12, 20, 46, 28, 41, 7, 30, 6, 52, 12, 38, 8, 34, 26, 58, 48, 60, 28, 22
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


COMMENTS

For no known n is a(n) = 1. If there is such an n it must be greater than 10^35 and have seven or more distinct prime factors (Hagis and Cohen 1982).  Jonathan Vos Post, May 01 2011
a(n) = 1 iff n is a power of 2. a(n) = 1  n iff n is prime.  Omar E. Pol, Jan 30 2014 [If a(n) = 1 then n is called a least deficient number or an almost perfect number. All the powers of 2 are least deficient numbers but it is not known if there exists a least deficient number that is not a power of 2. See A000079.  Jianing Song, Oct 13 2019]
According to Deléglise (1998), the abundant numbers have natural density 0.2474 < A(2) < 0.2480 (cf. A302991). Since the perfect numbers having density 0, the deficient numbers have density 0.7520 < 1  A(2) < 0.7526 (cf. A318172).  Daniel Forgues, Oct 10 2015
2abundance of n, a special case of the kabundance of n, defined as (sum of divisors of n)  k*n, k >= 1.  Daniel Forgues, Oct 24 2015


REFERENCES

Richard K. Guy, "Almost Perfect, QuasiPerfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers." Section B2 in Unsolved Problems in Number Theory, 2nd ed., New York: SpringerVerlag, pp. 4553, 1994.


LINKS

Eric Weisstein's World of Mathematics, Abundance.
Eric Weisstein's World of Mathematics, Abundancy.


FORMULA



EXAMPLE

For n = 10 the divisors of 10 are 1, 2, 5, 10. The sum of proper divisors of 10 minus 10 is 1 + 2 + 5  10 = 2, so the abundance of 10 is a(10) = 2.  Omar E. Pol, Dec 27 2013


MAPLE

with(numtheory); n>sigma(n)  2*n;


MATHEMATICA

Table[DivisorSigma[1, n]  2*n, {n, 1, 70}] (* Amiram Eldar, Jun 09 2022 *)


PROG



CROSSREFS



KEYWORD

sign,nice


AUTHOR



EXTENSIONS

Definition corrected Jul 04 2005


STATUS

approved



