A033880 - OEIS

%I #88 Apr 12 2024 18:59:36

%S -1,-1,-2,-1,-4,0,-6,-1,-5,-2,-10,4,-12,-4,-6,-1,-16,3,-18,2,-10,-8,

%T -22,12,-19,-10,-14,0,-28,12,-30,-1,-18,-14,-22,19,-36,-16,-22,10,-40,

%U 12,-42,-4,-12,-20,-46,28,-41,-7,-30,-6,-52,12,-38,8,-34,-26,-58,48,-60,-28,-22

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

%C 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

%C 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]

%C 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

%C 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

%C 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

%D 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.

%H J. G. Wurtzel, <a href="/A033880/b033880.txt">Table of n, a(n) for n = 1..10000</a> [This replaces an earlier b-file computed by T. D. Noe]

%H Nichole Davis, Dominic Klyve and Nicole Kraght, <a href="http://dx.doi.org/10.2140/involve.2013.6.493">On the difference between an integer and the sum of its proper divisors</a>, Involve, Vol. 6 (2013), No. 4, 493-504; DOI: 10.2140/involve.2013.6.493.

%H Marc Deléglise, <a href="http://projecteuclid.org/euclid.em/1048515661">Bounds for the density of abundant integers</a>, Experiment. Math. Volume 7, Issue 2 (1998), 137-143.

%H Peter Hagis Jr. and Graeme L. Cohen, <a href="http://dx.doi.org/10.1017/S1446788700018401">Some Results Concerning Quasiperfect Numbers</a>, J. Austral. Math. Soc. Ser. A 33, 275-286, 1982.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Abundance.html">Abundance</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Abundancy.html">Abundancy</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/QuasiperfectNumber.html">Quasiperfect Number</a>.

%F a(n) = A000203(n) - A005843(n). - _Omar E. Pol_, Dec 14 2008

%F a(n) = A001065(n) - n. - _Omar E. Pol_, Dec 27 2013

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

%e 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

%p with(numtheory); n->sigma(n) - 2*n;

%t Array[Total[Divisors[#]]-2#&,70] (* _Harvey P. Dale_, Sep 16 2011 *)

%t Table[DivisorSigma[1, n] - 2*n, {n, 1, 70}] (* _Amiram Eldar_, Jun 09 2022 *)

%o (PARI) a(n)=sigma(n)-2*n \\ _Charles R Greathouse IV_, Nov 20 2012

%o (Magma) [SumOfDivisors(n)-2*n: n in [1..100]]; // _Vincenzo Librandi_, Oct 11 2015

%o (Python)

%o from sympy import divisor_sigma

%o def A033880(n): return divisor_sigma(n)-(n<<1) # _Chai Wah Wu_, Apr 12 2024

%Y Equals -A033879.

%Y Cf. A000203, A001065. A005843.

%Y 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).

%Y Cf. A072691, A302991, A318172.

%K sign,nice

%O 1,3

%A _N. J. A. Sloane_

%E Definition corrected Jul 04 2005