A033880 Abundance of n, or (sum of divisors of n) - 2n.

-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

Offset: 1

Author

N. J. A. Sloane

Keywords

• sign
• nice

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

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

Examples

			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
		

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.
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.

Links

• J. G. Wurtzel, Table of n, a(n) for n = 1..10000 [This replaces an earlier b-file 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, 493-504; DOI: 10.2140/involve.2013.6.493.
• Marc Deléglise, Bounds for the density of abundant integers, Experiment. Math. Volume 7, Issue 2 (1998), 137-143.
• Peter Hagis Jr. and Graeme L. Cohen, Some Results Concerning Quasiperfect Numbers, J. Austral. Math. Soc. Ser. A 33, 275-286, 1982.
• Eric Weisstein's World of Mathematics, Abundance.
• Eric Weisstein's World of Mathematics, Abundancy.
• Eric Weisstein's World of Mathematics, Quasiperfect Number.

Crossrefs

Equals -A033879.

Cf. A000203, A001065. A005843.

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. A072691, A302991, A318172.

Programs

Magma

[SumOfDivisors(n)-2*n: n in [1..100]]; // Vincenzo Librandi, Oct 11 2015

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 *)

PARI

a(n)=sigma(n)-2*n \\ Charles R Greathouse IV, Nov 20 2012

Python

from sympy import divisor_sigma
def A033880(n): return divisor_sigma(n)-(n<<1) # Chai Wah Wu, Apr 12 2024

SageMath

[sigma(n, 1)-2*n for n in range(1, 64)] # Stefano Spezia, Jul 18 2025

Formula

a(n) = A000203(n) - A005843(n). - Omar E. Pol, Dec 14 2008

a(n) = A001065(n) - n. - Omar E. Pol, Dec 27 2013

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/12 - 1 = -0.1775329665... . - Amiram Eldar, Apr 06 2024

Extensions

Definition corrected Jul 04 2005