|
|
A033880
|
|
Abundance of n, or (sum of divisors of n) - 2n.
|
|
110
|
|
|
-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
2-abundance of n, a special case of the k-abundance of n, defined as (sum of divisors of n) - k*n, k >= 1. - Daniel Forgues, Oct 24 2015
|
|
REFERENCES
|
Richard K. Guy, "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers." Section B2 in Unsolved Problems in Number Theory, 2nd ed., New York: Springer-Verlag, pp. 45-53, 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
|
|
|
|