A number m is abundant if sigma(m) > 2m (this sequence), perfect if sigma(m) = 2m (cf. A000396), or deficient if sigma(m) < 2m (cf. A005100), where sigma(m) is the sum of the divisors of m (A000203).

While the first even abundant number is 12 = 2^2*3, the first odd abundant is 945 = 3^3*5*7, the 232nd abundant number!

If m is a term so is every positive multiple of m. "Primitive" terms are in A091191.

If m=6k (k>=2), then sigma(m) >= 1 + k + 2*k + 3*k + 6*k > 12*k = 2*m. Thus all such m are in the sequence.

According to Deléglise (1998), the abundant numbers have natural density 0.2474 < A(2) < 0.2480. Thus the n-th abundant number is asymptotic to 4.0322*n < n/A(2) < 4.0421*n. - Daniel Forgues, Oct 11 2015

From Bob Selcoe, Mar 28 2017 (prompted by correspondence with Peter Seymour): (Start)

Applying similar logic as the proof that all multiples of 6 >= 12 appear in the sequence, for all odd primes p:

i) all numbers of the form j*p*2^k (j >= 1) appear in the sequence when p < 2^(k+1) - 1;

ii) no numbers appear when p > 2^(k+1) - 1 (i.e., are deficient and are in A005100);

iii) when p = 2^(k+1) - 1 (i.e., perfect numbers, A000396), j*p*2^k (j >= 2) appear.

Note that redundancies are eliminated when evaluating p only in the interval [2^k, 2^(k+1)].

The first few even terms not of the forms i or iii are {70, 350, 490, 550, 572, 650, 770, ...}. (End)

L. E. Dickson, Theorems and tables on the sum of the divisors of a number, Quart. J. Pure Appl. Math., Vol. 44 (1913), pp. 264-296.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, pp. 74-84.

Ross Honsberger, Ingenuity in Mathematics, Random House, 1970, p. 125.

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 59.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 128.

S. Flora Jeba, Anirban Roy, and Manjil P. Saikia, On k-Facile Perfect Numbers, Algebra and Its Applications (ICAA-2023) Springer Proc. Math. Stat., Vol. 474, 111-121. See p. 112.

Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015. [A later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication below.]

Eric Weisstein's World of Mathematics, Abundance.

a(n) is asymptotic to C*n with C=4.038... (Deléglise, 1998). - Benoit Cloitre, Sep 04 2002

with(numtheory): for n from 1 to 270 do if sigma(n)>2*n then printf(`%d, `, n) fi: od:

isA005101 := proc(n)

simplify(numtheory[sigma](n) > 2*n) ;

option remember ;

local a ;

if n =1 then

12 ;

else

a := procname(n-1)+1 ;

while numtheory[sigma](a) <= 2*a do

a := a+1 ;

end do ;

a ;

end if ;

abQ[n_] := DivisorSigma[1, n] > 2n; A005101 = Select[ Range[270], abQ[ # ] &] (* Robert G. Wilson v, Sep 15 2005 *)

Select[Range[300], DivisorSigma[1, #] > 2 # &] (* Vincenzo Librandi, Oct 12 2015 *)

(PARI) isA005101(n) = (sigma(n) > 2*n) \\ Michael B. Porter, Nov 07 2009

(Haskell)

a005101 n = a005101_list !! (n-1)

a005101_list = filter (\x -> a001065 x > x) [1..]

(Python)

from sympy import divisors

def ok(n): return sum(divisors(n)) > 2*n

(Python)

from sympy import divisor_sigma

from itertools import count, islice

def A005101_gen(startvalue=1): return filter(lambda n:divisor_sigma(n) > 2*n, count(max(startvalue, 1))) # generator of terms >= startvalue

Cf. A094268 (n consecutive abundant numbers).

Cf. A173490 (even abundant numbers).

Cf. A000396 (perfect numbers).

nonn,easy,core,nice,changed