

A033880


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


107



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
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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
Not to be confused with the abundancy of n, defined as (sum of divisors of n) / n. (Cf. A017665 / A017666.)  Daniel Forgues, Oct 25 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

J. G. Wurtzel, Table of n, a(n) for n = 1..10000 [This replaces an earlier bfile computed by T. D. Noe]
Nichole Davis, Dominic Klyve and Nicole Kraght, On the difference between an integer and the sum of its proper divisors, Involve, Vol. 6 (2013), No. 4, 493504; DOI: 10.2140/involve.2013.6.493.
Marc Deléglise, Bounds for the density of abundant integers, Experiment. Math. Volume 7, Issue 2 (1998), 137143.
P. Hagis and G. L. Cohen, Some Results Concerning Quasiperfect Numbers, J. Austral. Math. Soc. Ser. A 33, 275286, 1982.
Eric Weisstein's World of Mathematics, Abundance.
Eric Weisstein's World of Mathematics, Abundancy.
Eric Weisstein's World of Mathematics, Quasiperfect Number.


FORMULA

a(n) = A000203(n)  A005843(n).  Omar E. Pol, Dec 14 2008
a(n) = A001065(n)  n.  Omar E. Pol, Dec 27 2013


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

Array[Total[Divisors[#]]2#&, 70] (* Harvey P. Dale, Sep 16 2011 *)
Table[DivisorSigma[1, n]  2*n, {n, 1, 70}] (* Amiram Eldar, Jun 09 2022 *)


PROG

(PARI) a(n)=sigma(n)2*n \\ Charles R Greathouse IV, Nov 20 2012
(Magma) [SumOfDivisors(n)2*n: n in [1..100]]; // Vincenzo Librandi, Oct 11 2015


CROSSREFS

Equals A033879.
a(n) = A000203(n)  A005843(n).  Omar E. Pol, Dec 14 2008
a(n) = A001065(n)  n.  Omar E. Pol, Dec 27 2013
Lists of positions where certain values occur: A005100 (a(n) < 0), A000396 (a(n) = 0) and A005101 (a(n) > 0), A023197 (a(n) >= n), A028982 (a(n) odd).
Cf. A302991, A318172.
Sequence in context: A233150 A103977 A109883 * A033879 A324546 A033883
Adjacent sequences: A033877 A033878 A033879 * A033881 A033882 A033883


KEYWORD

sign,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

Definition corrected Jul 04 2005


STATUS

approved



